{"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":1394,"title":"Prime Ladders","description":"A \u003chttp://en.wikipedia.org/wiki/Word_ladder word ladder\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\r\n\r\n COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\r\n\r\nA number ladder does much the same thing, changing one digit at a time. A *prime ladder* is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\r\n\r\n 757 \r\n 157\r\n 137\r\n 139\r\n\r\nGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element. \r\n\r\nTo restate the above example, consider\r\n\r\n p1 = 757\r\n p2 = 139\r\n\r\nfor which an acceptable answer is\r\n\r\n ladder = [757; 157; 137; 139]\r\n\r\nYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\r\n\r\n","description_html":"\u003cp\u003eA \u003ca href = \"http://en.wikipedia.org/wiki/Word_ladder\"\u003eword ladder\u003c/a\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/p\u003e\u003cpre\u003e COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\u003c/pre\u003e\u003cp\u003eA number ladder does much the same thing, changing one digit at a time. A \u003cb\u003eprime ladder\u003c/b\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/p\u003e\u003cpre\u003e 757 \r\n 157\r\n 137\r\n 139\u003c/pre\u003e\u003cp\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/p\u003e\u003cp\u003eTo restate the above example, consider\u003c/p\u003e\u003cpre\u003e p1 = 757\r\n p2 = 139\u003c/pre\u003e\u003cp\u003efor which an acceptable answer is\u003c/p\u003e\u003cpre\u003e ladder = [757; 157; 137; 139]\u003c/pre\u003e\u003cp\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/p\u003e","function_template":"function ladder = prime_ladder(p1,p2)\r\n  ladder = 0;\r\nend","test_suite":"%%\r\n\r\np1 = 13;\r\np2 = 29;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 389;\r\np2 = 269;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 761;\r\np2 = 397;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 983;\r\np2 = 239;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 271;\r\np2 = 439;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 877;\r\np2 = 733;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 2267;\r\np2 = 1153;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":3,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":"2013-03-27T21:24:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-26T22:51:16.000Z","updated_at":"2026-01-03T14:28:57.000Z","published_at":"2013-03-27T15:28:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Word_ladder\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eword ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ COLD\\n CORD\\n CARD\\n WARD\\n WARM]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number ladder does much the same thing, changing one digit at a time. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprime ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 757 \\n 157\\n 137\\n 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo restate the above example, consider\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p1 = 757\\n p2 = 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor which an acceptable answer is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ladder = [757; 157; 137; 139]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003c/w:body\u003e\u003c/w:document\u003e\"},{\"partUri\":\"/matlab/output.xml\",\"contentType\":\"text/xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\" standalone=\\\"no\\\" ?\u003e\u003cembeddedOutputs\u003e\u003cmetaData\u003e\u003cevaluationState\u003emanual\u003c/evaluationState\u003e\u003clayoutState\u003ecode\u003c/layoutState\u003e\u003coutputStatus\u003eready\u003c/outputStatus\u003e\u003c/metaData\u003e\u003coutputArray type=\\\"array\\\"/\u003e\u003cregionArray type=\\\"array\\\"/\u003e\u003c/embeddedOutputs\u003e\"}]}"}],"problem_search":{"errors":[],"problems":[{"id":1394,"title":"Prime Ladders","description":"A \u003chttp://en.wikipedia.org/wiki/Word_ladder word ladder\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\r\n\r\n COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\r\n\r\nA number ladder does much the same thing, changing one digit at a time. A *prime ladder* is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\r\n\r\n 757 \r\n 157\r\n 137\r\n 139\r\n\r\nGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element. \r\n\r\nTo restate the above example, consider\r\n\r\n p1 = 757\r\n p2 = 139\r\n\r\nfor which an acceptable answer is\r\n\r\n ladder = [757; 157; 137; 139]\r\n\r\nYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\r\n\r\n","description_html":"\u003cp\u003eA \u003ca href = \"http://en.wikipedia.org/wiki/Word_ladder\"\u003eword ladder\u003c/a\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/p\u003e\u003cpre\u003e COLD\r\n CORD\r\n CARD\r\n WARD\r\n WARM\u003c/pre\u003e\u003cp\u003eA number ladder does much the same thing, changing one digit at a time. A \u003cb\u003eprime ladder\u003c/b\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/p\u003e\u003cpre\u003e 757 \r\n 157\r\n 137\r\n 139\u003c/pre\u003e\u003cp\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/p\u003e\u003cp\u003eTo restate the above example, consider\u003c/p\u003e\u003cpre\u003e p1 = 757\r\n p2 = 139\u003c/pre\u003e\u003cp\u003efor which an acceptable answer is\u003c/p\u003e\u003cpre\u003e ladder = [757; 157; 137; 139]\u003c/pre\u003e\u003cp\u003eYou can assume that p1 and p2 contain the same number of digits. I am not looking for a unique answer. I will only check that the conditions of a prime ladder are met.\u003c/p\u003e","function_template":"function ladder = prime_ladder(p1,p2)\r\n  ladder = 0;\r\nend","test_suite":"%%\r\n\r\np1 = 13;\r\np2 = 29;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 389;\r\np2 = 269;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 761;\r\np2 = 397;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 983;\r\np2 = 239;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 271;\r\np2 = 439;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 877;\r\np2 = 733;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n\r\n%%\r\n\r\np1 = 2267;\r\np2 = 1153;\r\nladder = prime_ladder(p1,p2);\r\n\r\nassert(all(isprime(ladder)))\r\nassert(iscolumn(ladder))\r\nassert(ladder(1)==p1)\r\nassert(ladder(end)==p2)\r\nassert(all(sum(diff(num2str(ladder))~=0,2)==1))\r\n","published":true,"deleted":false,"likes_count":10,"comments_count":3,"created_by":7,"edited_by":null,"edited_at":null,"deleted_by":null,"deleted_at":null,"solvers_count":55,"test_suite_updated_at":"2013-03-27T21:24:26.000Z","rescore_all_solutions":false,"group_id":1,"created_at":"2013-03-26T22:51:16.000Z","updated_at":"2026-01-03T14:28:57.000Z","published_at":"2013-03-27T15:28:59.000Z","restored_at":null,"restored_by":null,"spam":false,"simulink":false,"admin_reviewed":false,"description_opc":"{\"relationships\":[{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/document\",\"targetMode\":\"\",\"relationshipId\":\"rId1\",\"target\":\"/matlab/document.xml\"},{\"relationshipType\":\"http://schemas.mathworks.com/matlab/code/2013/relationships/output\",\"targetMode\":\"\",\"relationshipId\":\"rId2\",\"target\":\"/matlab/output.xml\"}],\"parts\":[{\"partUri\":\"/matlab/document.xml\",\"relationship\":[],\"contentType\":\"application/vnd.mathworks.matlab.code.document+xml\",\"content\":\"\u003c?xml version=\\\"1.0\\\" encoding=\\\"UTF-8\\\"?\u003e\\n\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\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:hyperlink w:docLocation=\\\"http://en.wikipedia.org/wiki/Word_ladder\\\"\u003e\u003cw:r\u003e\u003cw:t\u003eword ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:hyperlink\u003e\u003cw:r\u003e\u003cw:t\u003e transforms one word to another by means of single-letter mutations. So COLD can become WARM like so (there are often multiple solutions):\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ COLD\\n CORD\\n CARD\\n WARD\\n WARM]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eA number ladder does much the same thing, changing one digit at a time. A\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e \u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:rPr\u003e\u003cw:b/\u003e\u003c/w:rPr\u003e\u003cw:t\u003eprime ladder\u003c/w:t\u003e\u003c/w:r\u003e\u003cw:r\u003e\u003cw:t\u003e is a number ladder with the additional constraint that each element is a prime number. Here is a prime ladder that connects 757 and 139\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ 757 \\n 157\\n 137\\n 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eGiven two numbers p1 and p2, construct a prime ladder column vector in which p1 is the first element, p2 is the last element, and each successive row differs by exactly one digit from the preceding element.\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eTo restate the above example, consider\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ p1 = 757\\n p2 = 139]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003efor which an acceptable answer is\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"code\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003e\u003c![CDATA[ ladder = [757; 157; 137; 139]]]\u003e\u003c/w:t\u003e\u003c/w:r\u003e\u003c/w:p\u003e\u003cw:p\u003e\u003cw:pPr\u003e\u003cw:pStyle w:val=\\\"text\\\"/\u003e\u003c/w:pPr\u003e\u003cw:r\u003e\u003cw:t\u003eYou can assume that p1 and p2 contain the same number of digits. 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