Maths Olympiad Prep

Track / Stage 4 / 326 of 340 #586 of 1964

Problem 586

AMC 12 late, AIME early
Number theory Difficulty 5.0 Find the answer

9.3. Find any five consecutive natural numbers less than 100, the product of which is divisible by 2014.

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

Official solution

Answer: 53, 54, 55, 56, 57.

This example can be obtained by noting that 2014=219532014=2 \cdot 19 \cdot 53.

Remark. This example of such numbers is unique. Only the answer without verification or explanation - 5 points; the answer with verification or explanation - 7 points.

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