Problem 30 · 2012 Math Kangaroo
Stretch
Number Theory
factorizationsum-constraint
Gerhard chooses two numbers a and b from the set {1, 2, 3, …, 26}. The product ab of these two numbers is equal to the sum of the remaining 24 numbers from this set. How big is |a − b|?
Show answer
Answer: E — 6
Show hints
Hint 1 of 2
Write 'product equals sum of the other 24' using the total \(1+2+\cdots+26 = 351\).
Still stuck? Show hint 2 →
Hint 2 of 2
Rearrange \(ab + a + b = 351\) into a product of two shifted factors (Simon's trick).
Show solution
Approach: turn the condition into a factoring of 352
- The remaining 24 numbers total \(351 - a - b\), so the condition is \(ab = 351 - a - b\).
- Add 1 to both sides: \(ab + a + b + 1 = 352\), i.e. \((a+1)(b+1) = 352\).
- The only factor pair of \(352 = 16\cdot22\) with both numbers \(\le 27\) gives \(\{a,b\} = \{15,21\}\), so \(|a-b| = 6\), choice E.
Mark:
· log in to save