在国内外的数学竞赛中,常有求整数p被整数q除所得余数或与之有关的问题,这类题目的解法往往因题而异,需要根据问题中数字的特征,灵活地应用某些技巧,因而常使选手们头痛。本文将应用下面介绍的整数除法中的几条简单性质,给出求解这类问题的一般思路,使问题按照既定的思路逐步化简,直至求出解答。以下为解题方便,我们把余数概念推广至可以是绝对值小于除数的负整数。设自然数P和q’被自然数d除时,余数 In mathematics competitions at home and abroad, there is often the problem of finding the remainder of the integer p divided by the integer q or the problems associated with it. The solutions to such problems often vary from question to question, depending on the characteristics of the numbers in the question, and some applications can be flexibly applied. Skills often cause headaches for players. This article will apply several simple properties of the integer division described below, and give the general idea of ​​solving such problems, so that the problem is gradually reduced according to the established thinking until the solution is obtained. The following is a convenient way to solve problems. We extend the concept of remainders to negative integers whose absolute value is less than the divisor. Let the natural numbers P and q’ be divided by the natural number d, the remainder