Competition problem — Number theory

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Number theory Difficulty 4.8 AIME Prove it United States

Problem:

Can the sum of three fourth powers end with the four digits 2019? (A fourth power is an integer of the form $n^{4}$ , where $n$ is an integer.)

Solution

Solution:

No, in fact it cannot even end in the digit 9. The possible last digits of a fourth power are $0, 1, 5, 6$ . No combination of three of these add up to 9.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.