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QUICKIE X (Posted on 2017-03-28) Difficulty: 1 of 5
Let x be a positive integer of the form 24n − 1, where n is an integer.

Prove that if a and b are positive integers such that x = ab,
then a + b is a multiple of 24.

See The Solution Submitted by Ady TZIDON    
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a bit sketchy but still a solution Comment 3 of 3 |
Odd squares are all = 1 mod 8
The squares of numbers not divisible by 3 are = 1 mod 3.

if ab = 24n - 1 then neither a nor b can be multiples of 2 or 3, since the RHS clearly isn't.

so ab = -1 mod 24 
We can safely reduce both a and b mod 24 to find a solution, and then consider:

a*(24-a) = 24a - a^2 =  -a^2 mod 24
but a^2 = 1 mod 8 and 1 mod 3 from the above, so a^2 = 1 mod 24 and (a, -a) is a solution mod 24. The sum of the two factors is indeed 0 mod 24

Adding multiples of 24 to either factor preserves the fact that the product = -1 mod 24, and preserves the fact that the sum = 0 mod 24

There aren't other solutions because if a*(24-a) = -1 mod 24 and a*(b) = -1 mod 24 then b = 24-a, since a has no factors in common with 24.  

  Posted by Paul on 2017-03-29 12:23:15
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