421 and 427, when divided by the same number, leave the same remainder 1. How many numbers can be used as the divisor in order to get the same remainder 1?

If both 421 and 427 leave remainder 1 when divided by N, then N divides (421−1)=420 and (427−1)=426. So N is a common divisor of 420 and 426. GCD(420, 426) = 6. Divisors of 6 are 1, 2, 3, 6. But N must be greater than the remainder 1, so N > 1. Hence N can be 2, 3, or 6 — that's 3 values.