Problem 17 · 2024 Math Kangaroo
Hard
Number Theory
factorizationgrouping
A number is written in each of the twelve circles shown. The number in each square is the product of the four numbers at the corners of that square. What is the product of the numbers in the eight bold circles?
Show answer
Answer: B — 40
Show hints
Hint 1 of 2
Multiply the four outer squares' numbers together and watch which circles get used and how many times.
Still stuck? Show hint 2 →
Hint 2 of 2
The centre square's number is the product of the four inner circles, and each inner circle sits in exactly two outer squares.
Show solution
Approach: multiply the four outer squares and divide out the inner circles
- Multiplying the four outer (arm) squares \(10\times4\times6\times24=5760\) uses each of the eight bold outer circles once and each of the four inner circles twice.
- The four inner circles multiply to the centre square's value 12, so their squared contribution is \(12^2=144\).
- Therefore the product of the eight bold circles is \(5760\div144=40\), answer B.
Mark:
· log in to save