Problem 158

Since 9 is a multiple of 3, then every positive integer that is a multiple of 9 is also a multiple of 3.

Therefore, we can simplify the problem to find the smallest positive integer that is a multiple of each of 5, 7 and 9.

The smallest positive integer that is a multiple of each of 7 and 9 is $7 \cdot 9 = 63$, since 7 and 9 have no common divisor larger than 1. (We could also list the positive multiples of 9 until we found the first one that is also a multiple of 7.)

Thus, the positive integers that are multiples of 7 and 9 are those which are multiples of 63.

We list the multiples of 63 until we find the first one that is divisible by 5 (that is, that ends in a 0 or in a 5): $$63 \cdot 1 = 63 \qquad 63 \cdot 2 = 126 \qquad 63 \cdot 3 = 189 \qquad 63 \cdot 4 = 252 \qquad 63 \cdot 5 = 315$$ Therefore, the smallest positive integer that is a multiple of each of 3, 5, 7, and 9 is 315.