Problem 25 · 2024 Math Kangaroo
Stretch
Number Theory
factorization
Dagobert wants to complete the diagram so that each box in the middle and top rows equals the product of the two boxes directly underneath it. Each box must contain a positive whole number, and he wants the top box to contain 36. How many different values can the number n have?
Show answer
Answer: D — 4
Show hints
Hint 1 of 2
Write the top box as a product of the three bottom boxes, with n appearing twice.
Still stuck? Show hint 2 →
Hint 2 of 2
The top equals the bottom product with an n-squared factor, so n-squared must divide 36.
Show solution
Approach: express the top as a product and require n-squared divides 36
- With bottom boxes x, n, y, the middle row is xn and ny, and the top is xn · ny = x·y·n² = 36.
- So n² must divide 36: n can be 1, 2, 3 or 6.
- Each choice leaves a valid positive product for x·y, so n has 4 possible values.
Mark:
· log in to save