Squares
Number Theory
Level
2
Rhona wrote down a list of nine multiples of ten: 10, 20, 30, 40, 50, 60, 70, 80, and 90. She then deleted some of the nine multiples so that the product of the remaining multiples was a perfect square. What is the least number of multiples that she could have deleted?
8
2
1
6
7
4
5
3
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1 solution
To end up with a square product, any prime factor must occur an even number of times. Rhona cannot use 70 since the factor 7 only appears once in 70, and never in any of the other numbers. If she uses all the other numbers she gets 10×20×30×40×50×60×80×90 = (2 × 5) × (2 × 2 × 5) × (2 × 3 × 5) × (2 × 2 × 2 × 5) × (2 × 5 × 5) ×(2×2×3×5)×(2×2×2×2×5)×(2×3×3×5) = 215 × 34 × 59. To get a square, she needs to make the powers of 2 and 5 even, say by removing 10 from the list (although removing 40 or 90 works too). Hence she needs to remove only two numbers from her list.