2023 AMC 10A Problem 3

P1-10~2 minPrint

Number Properties Basic Counting Primes Bound the quantity above/below or estimate to pin it down Set up the equation/formula and compute; no special trick needed

How many positive perfect squares less than $2023$ are divisible by $5$ ?

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Key insight

A square is a multiple of 5 only when the number squared is a multiple of 5. Count 5, 10, 15, and so on while the square stays below 2023.

Solution

First decide which squares are divisible by $5$ . Try a few: $5^2 = 25$ and $10^2 = 100$ are, but $6^2 = 36$ and $7^2 = 49$ are not. Since $5$ is prime, $5$ can only divide $n \times n$ if it divides $n$ itself. So we want the squares of $5, 10, 15, 20, \ldots$
Next find where the squares stop being less than $2023$ . We know $40^2 = 1600$ and $45^2 = 2025$ .
$2025$ is just above $2023$ , so $45$ is not allowed. The last one that works is $40$ .
List them: $5, 10, 15, 20, 25, 30, 35, 40$ . Their squares $25, 100, 225, 400, 625, 900, 1225, 1600$ are all below $2023$ and all divisible by $5$ .
That is $8$ squares.

The answer is $\boxed{\textbf{(A)}\ 8}$ .

Why this works

A prime number divides a square only when it divides the number being squared, so the question about squares turns into a question about multiples of $5$ . The only tricky part is the cutoff: $45^2 = 2025$ misses $2023$ by just $2$ , and that single case decides between (A) and (B). Always check the last case by hand.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Because $5$ is prime, $5$ divides $n^2$ exactly when $5$ divides $n$ . So we need to count positive multiples of $5$ whose squares are below $2023$ .

Since $44^2 = 1936 < 2023$ and $45^2 = 2025 > 2023$ , the roots allowed are $n = 1, 2, \ldots, 44$ . The multiples of $5$ in this range are $5, 10, 15, \ldots, 40$ , which is $8$ numbers.

Their squares are $25, 100, 225, \ldots, 1600$ , all below $2023$ and all divisible by $5$ .

The answer is $\boxed{\textbf{(A)}\ 8}$ .

Review in the AoPS books

Introduction to Number Theory, Chapter 1: Integers: The Basics
Introduction to Number Theory, Chapter 2: Primes and Composites
Introduction to Counting & Probability, Chapter 1: Counting Is Arithmetic
Introduction to Algebra, Chapter 20: Special Functions (§20.3)