2025 AMC 8 Problems/Problem 23
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
Note that if a perfect square ends in "", then when
is subtracted from this number, (Condition II) the number will end in "
" (Condition I). Therefore, the number is in the form
, where
(otherwise
won't end in "
" or
won't be
digits). Also, note that
. Therefore,
and
are both prime numbers because of (Condition III). Testing, we get
Out of these, the only number that is the product of prime numbers is
, so the answer is
. four-digit number
~Soupboy0