Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Find the answer

2. Given the sets
$A=\left\{n^{2}+1 \mid n \in \mathbf{Z}_{+}\right\}, B=\left\{n^{3}+1 \mid n \in \mathbf{Z}_{+}\right\} \text {. }$

Arrange all elements in $A \cap B$ in ascending order to form the sequence $a_{1}, a_{2}, \cdots$ . Then the units digit of $a_{99}$ is

Your answer

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

Solution

2. 2 .

From the given, we know that $A \cap B=\left\{n^{6}+1 \mid n \in \mathbf{Z}_{+}\right\}$ .
Therefore, $a_{99}=99^{6}+1$ .
Thus, its unit digit is 2.

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