Find the sum of the remainders obtained when a number n is divided by 9 and 7 successively, if n is the smallest number that leaves respective remainders of 4, 6 and 9 when divided successively by 13, 11 and 15.4596

The problem involves the Chinese Remainder Theorem. Here are the steps to solve it:

Step 1: We first need to find the smallest number n that leaves respective remainders of 4, 6 and 9 when divided successively by 13, 11 and 15.

We start with the last divisor. The number n is of the form 15k + 9 for some integer k. We substitute this into the second condition to get 15k + 9 = 11m + 6 for some integer m. Simplifying, we get 15k - 11m = -3. The smallest solution to this Diophantine equation is k = 7, m = 9.

Substituting k = 7 into n = 15k + 9, we get n = 15*7 + 9 = 114.

Step 2: Now we need to find the sum of the remainders obtained when n is divided by 9 and 7 successively.

When 114 is divided by 9, the remainder is 6. When 114 is divided by 7, the remainder is 4.

So, the sum of the remainders is 6 + 4 = 10.