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Number Theory Level 2

My watch shows the day's date in the form M M ⋅ D D ⋅ Y Y MM \cdot DD \cdot YY . For instance, the day I posted this problem, my watch read 01 ⋅ 05 ⋅ 17 01 \cdot 05 \cdot 17 .

How many times in 2017 will the product M M × D D × Y Y MM \times DD \times YY be a perfect square ?


The answer is 3.

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Guilherme Dela Corte by Guilherme Dela Corte , 24, Brazil

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1 solution

Since 17 17 is prime, the only way for M M × D D × 17 MM \times DD \times 17 to be a perfect square is to have D D = 17 DD = 17 .

This leaves us to M M MM being a perfect square too. We have M M = 1 , 4 MM = 1, \; 4 or 9 9 .

Thus, only three dates this year will lead M M × D D × Y Y MM \times DD \times YY to be a perfect square.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

This question will be more challenging if you give this question in 2016 i.e. DD=16

Kushal Bose - 4 years, 5 months ago

Nice problem.

Swapnil Das - 4 years, 5 months ago

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