A puzzle by S. Dvoryaninov from the July-August 1994 issue of Quantum:
A very large military band marched in square formation on a parade ground, then regrouped into a rectangle so that the number of rows increased by 5. How many musicians were in the band?
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This solution is by V. Dubrovsky. Suppose the original square formation measures n × n and contains n2 musicians. We know that n2 must be divisible by n + 5, because it’s possible to regroup the square formation into a rectangle with n + 5 rows. Since n2 = (n + 5)(n – 5) + 25, 25 must be divisible by n + 5. And since the only divisor of 25 that’s greater than 5 is 25 itself, n + 5 must be 25. Thus n = 20 and the number of musicians in the band is n2 = 400.
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