Problem 21 · 2013 Math Kangaroo
Stretch
Number Theory
factorization
How many pairs of integers \((x, y)\) with \(x \le y\) are there such that their product is exactly five times their sum?
Show answer
Answer: A — 4
Show hints
Hint 1 of 2
Rewrite xy = 5(x+y) so the variables separate.
Still stuck? Show hint 2 →
Hint 2 of 2
Add a constant to both sides to factor as a product equal to 25.
Show solution
Approach: Simon-style factoring
- xy = 5x + 5y → (x−5)(y−5) = 25.
- Integer factor pairs of 25 (with x ≤ y) give (6,30), (10,10), (0,0), (−20,4).
- That is 4 pairs: A.
Mark:
· log in to save