Problem 4 · 2020 Math Kangaroo
Medium
Number Theory
factor-pairscasework
Four integer numbers have sum S and product 9. What is the lowest possible value for S?
Show answer
Answer: B — −12
Show hints
Hint 1 of 2
The product is 9, so the four numbers come from the factors of 9 — but signs are free.
Still stuck? Show hint 2 →
Hint 2 of 2
To make the sum as small as possible, use negatives; an even number of them keeps the product positive.
Show solution
Approach: choose factors of 9 with an even count of negatives to minimise the sum
- Nine factors into magnitudes 9, 1, 1, 1 (or 3, 3, 1, 1).
- Take four negatives so the product stays positive: −9, −1, −1, −1.
- Their product is 9 and their sum is −12.
- No choice goes lower, so the smallest sum is −12.
Mark:
· log in to save