Track / Stage 4 / 250 of 340 #510 of 1964

Problem 510

AMC 12 late, AIME early

Number theory Difficulty 4.9 Find the answer

$1 \cdot 6$ Let $n$ be an integer. If the tens digit of $n^{2}$ is 7, what is the units digit of $n^{2}$ ?

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

[Solution] Let $n=10 x+y$ , where $x$ and $y$ are integers, and $0 \leqslant y \leqslant 9$ . Thus, we have
$\begin{aligned} n^{2} & =100 x^{2}+20 x y+y^{2} \\ & =20\left(5 x^{2}+x y\right)+y^{2} \end{aligned}$

If the tens digit of $n^{2}$ is the odd number 7, then the tens digit of $y^{2}$ is odd, which leads to
$y^{2}=16 \text { or } 36 \text {. }$

Therefore, the units digit of $n^{2}$ must be 6.

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