Solution #1

24 = 2^3 * 3, and 70 = 2 * 5 * 7.

We have 24x divisible by 70, so 2, 5, and 7 all need to be factors of 24x. 2 is already a factor of 24, but 5 and 7 are not. So 5 and 7 need to be factors of x.

We have 70x divisible by 24, so 2^3 and 3 need to be factors of 70x. 2 is already a factor of 70, so we need an additional 2^2 and a 3 in x.

So x needs to contain factors of 2^2, 3, 5, and 7. To minimize x, we use just these factors. Thus x = 2^2 * 3 * 5 * 7 = 420.