Maths Olympiad Prep

Track / Stage 4 / 143 of 340 #403 of 1964

Problem 403

AMC 12 late, AIME early
Number theory Difficulty 4.8 Find the answer

An Unusual Division Example. Here is a rather curious puzzle. Find the smallest number that, when successively divided by 45,454,454545, 454, 4545, and 45454, gives remainders of 4,45,4544, 45, 454, and 4545, respectively. Finding such a number might not be easy, but solving the problem will refresh your knowledge of arithmetic.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

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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 reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.