Difference between revisions of "2025 AMC 8 Problems/Problem 23"
| Line 38: | Line 38: | ||
https://youtu.be/jn-qIwv57nQ | https://youtu.be/jn-qIwv57nQ | ||
| − | ==A Complete Video Solution with the Thought Process by Dr. Xue's Math School | + | ==A Complete Video Solution with the Thought Process by Dr. Xue's Math School== |
https://youtu.be/aEPPwMIQ52w | https://youtu.be/aEPPwMIQ52w | ||
Latest revision as of 02:16, 3 February 2025
Contents
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.
Solution
The Condition II perfect square must end in "
" because 
(Condition I). Four-digit perfect squares ending in "" are 
.
Condition II also says the number is in the form 
. By Difference of Squares[1], 
. So:
On this list, the only number that is the product of 
prime numbers is 
, so the answer is 
.
~Soupboy0
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=jTTcscvcQmI
Video Solution (A Clever Explanation You’ll Get Instantly)
https://youtu.be/VP7g-s8akMY?si=wexxSYnEz2IcjeIb&t=3539 ~hsnacademy
Video Solution by Thinking Feet
Video Solution by Dr. David
A Complete Video Solution with the Thought Process by Dr. Xue's Math School
See Also
| 2025 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 22 |
Followed by Problem 24 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
