2019 AIME II Problems/Problem 3
Find the number of -tuples of positive integers that satisfy the following systems of equations:
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 .