2025 AMC 8 Problem 23
Attempt Problem 23 of the 2025 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
23.
How many four-digit numbers have all three of the following properties?
(I) The tens digit and ones digit are both
(II) The number is less than a perfect square.
(III) The number is the product of exactly two prime numbers.
Answer: B
Small hint:
The square one larger must end in
Big hint:
Check multiples of whose two neighbors are prime
Video solution:
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Written solution:
The number has the form and so the perfect square that is more ends in and so it is the square of a number ending in Suppose that square is
In order for to be a -digit number ending in , the only possibilities for are
Since we also need both and to be prime. We are then looking for pairs of prime numbers that are right around
Going through all the possibilities, the only ones that work are and , and so there is exactly way to do this. The answer is B.
Problem 23 in other years
- 1985 AMC 8
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