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Number theory Difficulty 4.3 AIME Find the answer United States

Problem:

In 2019, a team, including professor Andrew Sutherland of MIT, found three cubes of integers which sum to $42$ :
$42 = \left(-8053873881207597\_\right)^3 + (80435758145817515)^3 + (12602123297335631)^3$
One of the digits, labeled by an underscore, is missing. What is that digit?

Your answer

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

Solution

Solution:

Let the missing digit be $x$ . Then, taking the equation modulo $10$ , we see that $2 \equiv -x^{3} + 5^{3} + 1^{3}$ . This simplifies to $x^{3} \equiv 4 \pmod{10}$ , which gives a unique solution of $x = 4$ .

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