2025 AMC 8 Problems/Problem 23

Problem

How many four-digit numbers have all three of the following properties?

(I) The tens and ones digit are both 9.

(II) The number is 1 less than a perfect square.

(III) The number is the product of exactly two prime numbers.

$\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

Solution

Note that if a perfect square ends in " $00$ ", then when $1$ is subtracted from this number, (Condition II) the number will end in " $99$ " (Condition I). Therefore, the number is in the form $n^2-1$ , where $n = \{40, 50, 60, 70, 80, 90\}$ (otherwise $n^2-1$ won't end in " $99$ " or $n$ won't be $4$ digits). Also, note that $n^2-1 = (n+1)(n-1)$ . Therefore, $n-1$ and $n+1$ are both prime numbers because of (Condition III). Testing, we get

$40^2-1 = (39)(41)$

$50^2-1 = (49)(51)$

$60^2-1 = (59)(61)$

$70^2-1 = (69)(71)$

$80^2-1 = (79)(81)$

$90^2-1 = (89)(91)$

Out of these, the only number that is the product of $2$ prime numbers is $60^2-1 = (59)(61)$ , so the answer is $\boxed{\text{(B)\ 1}}$ . four-digit number

~Soupboy0

Vide Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=jTTcscvcQmI

Video Solution by Thinking Feet

https://youtu.be/PKMpTS6b988

2025 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2025 AMC 8 Problems/Problem 23

Contents

Problem

Solution

Vide Solution 1 by SpreadTheMathLove

Video Solution by Thinking Feet

See Also