For any real constants λ1, λ2 C （0, 1], let n ≥ max{[1/λ1 ], [1/λ2]}, vn ≥ 2 be integers. Suppose integers a C [1, λ1n] and b E [1, λ2n] satisfy the congruence b ≡ am （rood n）. The main purpose of this paper is to study the mean value of （a - b）2k for any fixed positive integer k and obtain some sharp asymptotic formulae.
