Question

what is the smallest integer that has remainder 2 when it is divided by 3,4,5,6,7?

Answer 1

This means, you find the smallest integer that is divisible by all these numbers and add 2 to it.That is LCM(3,4,5,6,7) +2.
LCM(3,4,5,6,7) =
1260
So your answer is 1262.

Answer 2

It turns out that there is a much smaller number that fits this description. Using the Chinese remainder theorem, the answer is actually 422.

Answer 3

My bad, the LCM of 3,4,5,6,7 is indeed 420!