There is more than one integer, greater than 1, which leaves a remainder of 1 when divided by each of the four smallest primes. What is the difference between the two smallest such integers?

210.

Step-by-step explanation:

The first 4 primes are 2,3,5 and 7.

If we multiply them we get 2*3*5*7 = 210.

So 210 + 1 = 211 is one integer which gives 1 as a remainder.

The next one would be 2*210 + 1 = 421.

The difference is 210.