Track / Stage 4 / 127 of 340 #387 of 1964

Problem 387

AMC 12 late, AIME early

Number theory Difficulty 4.8 Find the answer

120. An Unusual Division Example. Here is a rather curious puzzle. Find the smallest number that, when successively divided by $45, 454, 4545$ , and 45454, gives remainders of $4, 45, 454$ , and 4545, respectively. Finding such a number might not be easy, but solving the problem will refresh your knowledge of arithmetic.

Your answer

A number or a short expression. Spacing, $ signs and \frac vs / are all fine.

Official solution

120. The smallest number that satisfies all the conditions is 35641667749. Other numbers are obtained by adding to this any integer multiple of 46895573610.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.