Question 887599: What is the number which is divided by 13,15,19 simultaneously and remainders are 1,2,3 respectively
Answer by robertb(5830)   (Show Source):
You can put this solution on YOUR website! Stated in modulo arithmetic, we have
n = 1 mod 13 (1)
n = 2 mod 15 (2)
n = 3 mod 19 (3)
since 13, 15, and 19 are pairwise relatively prime, by the Chinese remainder theorem there is a set of solutions for this simultaneous system.
(1) says that n = 1+13q. Putting this into (2),
1 + 13q = 2 mod 15, or
13q = 1 mod 15, or  mod 15, or
q = 7 + 15l
=> n = 1 + 13(7 + 15l), or
n = 92 + 195l. Put this into (3):
92 + 195l = 3 mod 19, or 195l = -89 mod 19, or
195l = 6 mod 19
=> 5l = 6 mod 19
One solution for l is 5, and so l = 5, 24, 43,..., or l = 5 mod 19
or l = 5 + 19k.
=> n = 92 + 195(5 + 19k) = 92 + 975 + 3705k, or
n = 1067 + 3705k.
Some of the numbers satisfying the problem are 1067, 4772, 8477, etc.
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