What is the smallest positive integer that is divisible by each of the integers from 3 to 7, inclusive

The integers from 3 to 7, inclusive are:

{3, 4, 5, 6, 7}

these numbers can be written as multiplication of prime factors as :

{3, 2*2, 5, 2*3, 7}

The smallest common multiple of these numbers must have factors which divide all the factors of these 5 numbers.

So the multiple must have these factors: 3, 2*2 (one two is not enough to divide 4), 5 and 7. We do not need to write 2 and three again, because we already have them.

so the smallest common multiple is 3*2*2*5*7=420

420