SOLUTION: What is the number which is divided by 13,15,19 simultaneously and remainders are 1,2,3 respectively

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.
RELATED QUESTIONS
When a number is divided by 3 and 5 the remainders are 2 and 1 respectively. The sum of... (answered by dfrazzetto)

When a certain number is divided by 2, there is a remainder of 1. When the same number is (answered by CubeyThePenguin,ikleyn)

Six consecutive positive integers are each divided by 6 and their remainders are... (answered by ikleyn)

the remainders when f(x)=x�+ax�+bx+c is divided by (x-1),(x+2) and (x-2) are respectively (answered by ikleyn)

When polynomial P(x) is divided by x + 1, x + 2, and x + 3, the remainders are 2, 3, and... (answered by Edwin McCravy,ikleyn)

1.)When the expression x^3+mx^2+nx-5 is divided by x+2 and x+3, the remainders are 2 and... (answered by KMST)

If a no. Is divided by 3,7&11 the remainders are 1,6&5 respectively. Then find the... (answered by ankor@dixie-net.com)

What least number, when divided by 26,36 and 56, leaves the remainders 6,16,36... (answered by Edwin McCravy)

How do I solve this following question: The remainders whe ax^4+bx cube - x - 7 is... (answered by robertb)