Does there exist a perfect square whose decimal representation ends with 2012?

## - Solution -
If \( a \) is an integer such that the decimal representation of \( a^2 \) ends with 2012, then \( x^2 \equiv 2 \pmod{10} \). However, it is easily verified that a square cannot be congruent to 2 modulo 10. Therefore, the answer is: no, such an integer does not exist.

## 2 Essentials (2nd-1st year) : Arithmetic

We refer to the arithmetic course available on the Animath website for details and complements (see the links at the top of page 14).

## 1 First Course/TD

Program: Integers, integer part, divisibility, Euclidean division, base \( b \) decomposition, gcd, lcm, Euclidean algorithm, Bézout's theorem, Gauss's lemma. For the detailed course, see the Animath Arithmetic poly.

## - Statements of exercises covered in class -