{"group":{"id":1,"name":"Community","lockable":false,"created_at":"2012-01-18T18:02:15.000Z","updated_at":"2025-12-14T01:33:56.000Z","description":"Problems submitted by members of the MATLAB Central community.","is_default":true,"created_by":161519,"badge_id":null,"featured":false,"trending":false,"solution_count_in_trending_period":0,"trending_last_calculated":"2025-12-14T00:00:00.000Z","image_id":null,"published":true,"community_created":false,"status_id":2,"is_default_group_for_player":false,"deleted_by":null,"deleted_at":null,"restored_by":null,"restored_at":null,"description_opc":null,"description_html":null,"published_at":null},"problems":[{"id":60561,"title":"Determine whether a number is a Blum integer","description":"A Blum integer is a semiprime—that is, the product of two distinct primes—whose factors have the form  for some integer . The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form  with  and . The number 123 is a semiprime because it is , but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \r\nWrite a function to determine whether the numbers in a matrix are Blum integers. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.5px; transform-origin: 407px 67.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.60833px 8px; transform-origin: 6.60833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Blum_integer\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eBlum integer\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.775px 8px; transform-origin: 14.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/55275\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003esemiprime\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 228.683px 8px; transform-origin: 228.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e—that is, the product of two distinct primes—whose factors have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"4k+3\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.1083px 8px; transform-origin: 31.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for some integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 273.808px 8px; transform-origin: 273.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"4k+3\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 8px; transform-origin: 16.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" alt=\"k = 0\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" alt=\"k = 1\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.083px 8px; transform-origin: 145.083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 123 is a semiprime because it is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"3x41\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 188.633px 8px; transform-origin: 188.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251.917px 8px; transform-origin: 251.917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether the numbers in a matrix are Blum integers. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isBlum(x)\r\n  y = false;\r\nend","test_suite":"%%\r\nx = 21;\r\nassert(isBlum(x))\r\n\r\n%%\r\nx = 25;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 123;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 123;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 737;\r\nassert(isBlum(x))\r\n\r\n%%\r\nx = 1463;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = [37533 37581 37601 37617 37641 37669 37709 37721 37749 37753 37769];\r\nassert(all(isBlum(x)))\r\n\r\n%%\r\nx = 4+[37533 37581 37601 37617 37641 37669 37709 37721 37749 37753 37769];\r\nassert(sum(isBlum(x))==1)\r\n\r\n%% \r\nx = [74089  74301  74401  74549  74669  74773  74881\r\n     74141  74309  74417  74553  74689  74801  74901\r\n     74213  74329  74429  74557  74701  74809  74913\r\n     74233  74369  74461  74569  74721  74813  74917\r\n     74249  74389  74473  74577  74741  74829  74937\r\n     74281  74393  74537  74621  74757  74833  74957\r\n     74289  74397  74541  74641  74769  74837  74977];\r\nassert(isequal(isBlum(x),true(7)))\r\n\r\n%% \r\nx = [74089  74301  74401  74549  74669  74773  74881\r\n     74141  74309  74417  74553  74689  74801  74901\r\n     74213  74329  74429  74557  74701  74809  74913\r\n     74233  74369  74461  74569  74721  74813  74917\r\n     74249  74389  74473  74577  74741  74829  74937\r\n     74281  74393  74537  74621  74757  74833  74957\r\n     74289  74397  74541  74641  74769  74837  74977];\r\nA = isBlum(x+16);\r\ny = A*2.^(6:-1:0)';\r\ny_correct = [16 9 0 66 4 20 17]';\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = randi(1000);\r\nx = 2*randi(450000,1,n);\r\nassert(all(~isBlum(x)))\r\n\r\n%%\r\nx = reshape([97091 106789 109033 147703 186819 267349 345133 472647 480241 491283 510839 524249 87893 128567 185019 226173 273351 325969 377121 405283 440487 478509 506273 518377],[4 3 2]);\r\ny = isBlum(x);\r\ny_correct = reshape(logical([0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1]),[4 3 2]);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [504737 539969 619369];\r\nassert(isequal(isBlum(x),logical([1 1 0])))\r\n\r\n%%\r\nf = 4*(1:200)+3;\r\nf1 = f(isprime(f));\r\nassert(isequal(isBlum(f1.*f1'),logical(~eye(length(f1)))))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-06-26T00:46:58.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-06-25T03:32:30.000Z","updated_at":"2025-07-31T19:37:37.000Z","published_at":"2024-06-25T03:33:13.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Blum_integer\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBlum integer\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/55275\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esemiprime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e—that is, the product of two distinct primes—whose factors have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"4k+3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4k+3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for some integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\\n\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"4k+3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4k+3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The number 123 is a semiprime because it is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"3x41\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e3\\\\times41\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine whether the numbers in a matrix are Blum integers. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\",\"relationship\":null}],\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"target\":\"/matlab/document.xml\",\"relationshipId\":\"rId1\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":60561,"title":"Determine whether a number is a Blum integer","description":"A Blum integer is a semiprime—that is, the product of two distinct primes—whose factors have the form  for some integer . The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form  with  and . The number 123 is a semiprime because it is , but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \r\nWrite a function to determine whether the numbers in a matrix are Blum integers. ","description_html":"\u003cdiv style = \"text-align: start; line-height: 20.4333px; min-height: 0px; white-space: normal; color: rgb(0, 0, 0); font-family: Menlo, Monaco, Consolas, monospace; font-style: normal; font-size: 14px; font-weight: 400; text-decoration: rgb(0, 0, 0); white-space: normal; \"\u003e\u003cdiv style=\"block-size: 135px; display: block; min-width: 0px; padding-block-start: 0px; padding-top: 0px; perspective-origin: 407px 67.5px; transform-origin: 407px 67.5px; vertical-align: baseline; \"\u003e\u003cdiv style=\"block-size: 105px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 52.5px; text-align: left; transform-origin: 384px 52.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 6.60833px 8px; transform-origin: 6.60833px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eA \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://en.wikipedia.org/wiki/Blum_integer\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003eBlum integer\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 14.775px 8px; transform-origin: 14.775px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e is a \u003c/span\u003e\u003c/span\u003e\u003ca target='_blank' href = \"https://www.mathworks.com/matlabcentral/cody/problems/55275\"\u003e\u003cspan style=\"\"\u003e\u003cspan style=\"text-decoration-line: underline; \"\u003esemiprime\u003c/span\u003e\u003c/span\u003e\u003c/a\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 228.683px 8px; transform-origin: 228.683px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e—that is, the product of two distinct primes—whose factors have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"4k+3\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 31.1083px 8px; transform-origin: 31.1083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e for some integer \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"font-family: \u0026quot;STIXGeneral\u0026quot;, \u0026quot;STIXGeneral-webfont\u0026quot;, serif; font-style: italic; font-weight: 400; color: rgb(0, 0, 0);\"\u003ek\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 273.808px 8px; transform-origin: 273.808px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"4k+3\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 16.3333px 8px; transform-origin: 16.3333px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e with \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" alt=\"k = 0\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 15.5583px 8px; transform-origin: 15.5583px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e and \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"36\" height=\"18\" alt=\"k = 1\" style=\"width: 36px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 145.083px 8px; transform-origin: 145.083px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e. The number 123 is a semiprime because it is \u003c/span\u003e\u003c/span\u003e\u003cspan style=\"vertical-align:-5px\"\u003e\u003cimg src=\"data:image/png;base64,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\" width=\"42\" height=\"18\" alt=\"3x41\" style=\"width: 42px; height: 18px;\"\u003e\u003c/span\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 188.633px 8px; transform-origin: 188.633px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003e, but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003cdiv style=\"block-size: 21px; font-family: Helvetica, Arial, sans-serif; line-height: 21px; margin-block-end: 9px; margin-block-start: 2px; margin-bottom: 9px; margin-inline-end: 10px; margin-inline-start: 4px; margin-left: 4px; margin-right: 10px; margin-top: 2px; perspective-origin: 384px 10.5px; text-align: left; transform-origin: 384px 10.5px; white-space-collapse: preserve; margin-left: 4px; margin-top: 2px; margin-bottom: 9px; margin-right: 10px; \"\u003e\u003cspan style=\"block-size: auto; display: inline; margin-block-end: 0px; margin-block-start: 0px; margin-bottom: 0px; margin-inline-end: 0px; margin-inline-start: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px; perspective-origin: 251.917px 8px; transform-origin: 251.917px 8px; unicode-bidi: normal; \"\u003e\u003cspan style=\"\"\u003eWrite a function to determine whether the numbers in a matrix are Blum integers. \u003c/span\u003e\u003c/span\u003e\u003c/div\u003e\u003c/div\u003e\u003c/div\u003e","function_template":"function tf = isBlum(x)\r\n  y = false;\r\nend","test_suite":"%%\r\nx = 21;\r\nassert(isBlum(x))\r\n\r\n%%\r\nx = 25;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 123;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 123;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = 737;\r\nassert(isBlum(x))\r\n\r\n%%\r\nx = 1463;\r\nassert(~isBlum(x))\r\n\r\n%%\r\nx = [37533 37581 37601 37617 37641 37669 37709 37721 37749 37753 37769];\r\nassert(all(isBlum(x)))\r\n\r\n%%\r\nx = 4+[37533 37581 37601 37617 37641 37669 37709 37721 37749 37753 37769];\r\nassert(sum(isBlum(x))==1)\r\n\r\n%% \r\nx = [74089  74301  74401  74549  74669  74773  74881\r\n     74141  74309  74417  74553  74689  74801  74901\r\n     74213  74329  74429  74557  74701  74809  74913\r\n     74233  74369  74461  74569  74721  74813  74917\r\n     74249  74389  74473  74577  74741  74829  74937\r\n     74281  74393  74537  74621  74757  74833  74957\r\n     74289  74397  74541  74641  74769  74837  74977];\r\nassert(isequal(isBlum(x),true(7)))\r\n\r\n%% \r\nx = [74089  74301  74401  74549  74669  74773  74881\r\n     74141  74309  74417  74553  74689  74801  74901\r\n     74213  74329  74429  74557  74701  74809  74913\r\n     74233  74369  74461  74569  74721  74813  74917\r\n     74249  74389  74473  74577  74741  74829  74937\r\n     74281  74393  74537  74621  74757  74833  74957\r\n     74289  74397  74541  74641  74769  74837  74977];\r\nA = isBlum(x+16);\r\ny = A*2.^(6:-1:0)';\r\ny_correct = [16 9 0 66 4 20 17]';\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nn = randi(1000);\r\nx = 2*randi(450000,1,n);\r\nassert(all(~isBlum(x)))\r\n\r\n%%\r\nx = reshape([97091 106789 109033 147703 186819 267349 345133 472647 480241 491283 510839 524249 87893 128567 185019 226173 273351 325969 377121 405283 440487 478509 506273 518377],[4 3 2]);\r\ny = isBlum(x);\r\ny_correct = reshape(logical([0 1 1 0 0 1 0 0 1 0 0 1 0 0 0 1 0 1 1 0 0 1 1 1]),[4 3 2]);\r\nassert(isequal(y,y_correct))\r\n\r\n%%\r\nx = [504737 539969 619369];\r\nassert(isequal(isBlum(x),logical([1 1 0])))\r\n\r\n%%\r\nf = 4*(1:200)+3;\r\nf1 = f(isprime(f));\r\nassert(isequal(isBlum(f1.*f1'),logical(~eye(length(f1)))))","published":true,"deleted":false,"likes_count":0,"comments_count":0,"created_by":46909,"edited_by":46909,"edited_at":"2024-06-26T00:46:58.000Z","deleted_by":null,"deleted_at":null,"solvers_count":8,"test_suite_updated_at":null,"rescore_all_solutions":false,"group_id":1,"created_at":"2024-06-25T03:32:30.000Z","updated_at":"2025-07-31T19:37:37.000Z","published_at":"2024-06-25T03:33:13.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\u003cw:document xmlns:w=\\\"http://schemas.openxmlformats.org/wordprocessingml/2006/main\\\"\u003e\u003cw:body\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://en.wikipedia.org/wiki/Blum_integer\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eBlum integer\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e is a \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"https://www.mathworks.com/matlabcentral/cody/problems/55275\\\"\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:u/\u003e\u003c/w:rPr\u003e\u003cw:t\u003esemiprime\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e—that is, the product of two distinct primes—whose factors have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"4k+3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4k+3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e for some integer \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek\\n\\n\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The number 21 is a Blum integer because its two prime factors, 3 and 7, have the form \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"4k+3\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e4k+3\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e with \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k = 0\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek = 0\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e and \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"k = 1\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003ek = 1\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e. The number 123 is a semiprime because it is \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:customXml w:element=\\\"equation\\\"\u003e\u003cw:customXmlPr\u003e\u003cw:attr w:name=\\\"displayStyle\\\" w:val=\\\"false\\\"/\u003e\u003cw:attr w:name=\\\"altTextString\\\" w:val=\\\"3x41\\\"/\u003e\u003c/w:customXmlPr\u003e\u003cw:r\u003e\u003cw:t\u003e3\\\\times41\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:customXml\u003e\u003cw:r\u003e\u003cw:t\u003e, but it is not a Blum integer because 41 is not 3 more than a multiple of 4. The number 1463 is not a Blum integer: although its prime factors are distinct and each 3 more than a multiple of 4, there are three of them, not two. \u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003cw:jc w:val=\\\"left\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eWrite a function to determine whether the numbers in a matrix are Blum integers. 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