How many sums?

Probability Level pending

Let x , y x,y and z z be not necessarily distinct positive integers satisfying x y z = 64 xyz = 64 , find the number of distinct values of A = x + y + z A=x+y+z .


The answer is 7.

Denton Young by Denton Young , 47, USA

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1 solution

Denton Young
May 29, 2016

The order of the numbers is irrelevant (if x = 4, y = 8, z = 2, A will be the same as if x = 2, y = 4, z = 8), so without loss of generality we can take x, y, z to be in increasing order. There are 7 possible distributions:

(4, 4, 4)

(2, 4, 8)

(2, 2, 16)

(1, 8, 8)

(1, 4, 16)

(1, 2, 32)

(1, 1, 64)

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Moderator note:

What is the best way to ensure that we have found all the possible arrangements of ( x , y , z ) (x, y, z) ?

Do what I did. First do the arrangements where the smallest term is 4, then where the smallest term is 2, then where the smallest term is 1, and systematically go through the middle terms from largest to smallest.

Denton Young - 5 years ago

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