Hi Yogendra,

The smallest integer that has a remainder of 1 when divided by 5 is 6. The next such integer is 6 + 5 = 11and the next is 11 + 5 = 16. Hence the list of integers that have a remainder of 1 when divided by 5 is

\[ 6, 11, 16, \cdot \cdot \cdot\]

The smallest integer that has a remainder of 2 when divided by 7 is 9. What is the list of integers that have a remainder of 2 when divided by 7? I'm looking for the smallest number that appears in both lists. It comes quickly, it is 16.

This is not the answer to your question as the remainder when 16 is divided by 9 is 7.

16 is the smallest integer that gives a remainder of 1 when divided by 5 and a remainder of 2 when divided by 7. What is the next integer that gives a remainder of 1 when divided by 5 and a remainder of 2 when divided by 7? You need to add a multiple of 5 and a multiple of 7. The smallest number that is a multiple of 5 and a multiple of 7 is $5 \times 7 = 35.$ Thus the next integer that gives a remainder of 1 when divided by 5 and a remainder of 2 when divided by 7 is $16 + 35 = 51.$ If you divide 51 by 9 is its remainder 3? No it's not. What is the next integer that gives a remainder of 1 when divided by 5 and a remainder of 2 when divided by 7? What is its remainder when divided by 9?

Continue until you find an answer to your question. Once you have an answer notice that this answer plus $5 \times 7 \times 9$ is another answer and that second answer plus $5 \times 7 \times 9$ is another answer, and so on. There are infinitely many correct answers to your question.

I hope this helps,
Penny