document.write( "Question 887599: What is the number which is divided by 13,15,19 simultaneously and remainders are 1,2,3 respectively \n" ); document.write( "

Algebra.Com's Answer #536762 by robertb(5830) $\"\"$ $\"About$

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Stated in modulo arithmetic, we have
\n" ); document.write( "n = 1 mod 13 (1)
\n" ); document.write( "n = 2 mod 15 (2)
\n" ); document.write( "n = 3 mod 19 (3)
\n" ); document.write( "since 13, 15, and 19 are pairwise relatively prime, by the Chinese remainder theorem there is a set of solutions for this simultaneous system.
\n" ); document.write( "(1) says that n = 1+13q. Putting this into (2),
\n" ); document.write( "1 + 13q = 2 mod 15, or
\n" ); document.write( "13q = 1 mod 15, or $\"q+=+13%5E-1+=+7+\"$ mod 15, or
\n" ); document.write( "q = 7 + 15l
\n" ); document.write( "=> n = 1 + 13(7 + 15l), or
\n" ); document.write( "n = 92 + 195l. Put this into (3):
\n" ); document.write( "92 + 195l = 3 mod 19, or 195l = -89 mod 19, or
\n" ); document.write( "195l = 6 mod 19
\n" ); document.write( "=> 5l = 6 mod 19
\n" ); document.write( "One solution for l is 5, and so l = 5, 24, 43,..., or l = 5 mod 19
\n" ); document.write( "or l = 5 + 19k.
\n" ); document.write( "=> n = 92 + 195(5 + 19k) = 92 + 975 + 3705k, or
\n" ); document.write( "n = 1067 + 3705k.
\n" ); document.write( "Some of the numbers satisfying the problem are 1067, 4772, 8477, etc. \n" ); document.write( "

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