Start by finding a number that 2,3,4,5, and 6 go into, then we'll add 1.

The Least Common Multiple of 2 & 3 is 6, so it must be a multiple of 6.

The Least Common Multiple of 6 & 4 is 12, so it must be a multiple of 12.

The Least Common Multiple of 12 & 5 is 60, so it must be a multiple of 60.

The Least Common Multiple of 60 & 6 is 60, so it must be a multiple of 60.

So we'll start checking numbers that are 1 more than a multiple of 60, and see what 7 goes into evenly.

61 leaves remainder 5.

121 leaves remainder 2.

181 leaves remainder 6.

241 leaves remainder 3.

301 leaves remainder 0.

Yay! The answer is 301! (Exclamation mark is for excitement, not factorial.)