2019 AIME II Problems/Problem 3
Problem 3
Find the number of -tuples of positive integers
that satisfy the following systems of equations:
Solution
As 71 is prime, ,
, and
must be 1, 1, and 71 (up to ordering). However, since
and
are divisors of 70 and 72 respectively, the only possibility is
. Now we are left with finding the number of solutions
satisfying
and
, which separates easily into two subproblems. The number of positive integer solutions to
simply equals the number of divisors of 70 (as we can choose a divisor for
, which uniquely determines
). As
, we have
solutions. Similarly,
, so
.
Then the answer is simply .
-scrabbler94