Find
all sets of three positive integers
whose product is seven times their sum.
[SECRET]
Q:일곱을 뜻하는 영어단어는?
A:seven
[solution]
Let x, y, z be the integers in increasing order.
(We will use LE for "less than or equal to".)
Then xyz=7(x+y+z) LE 21z, so that xy LE 21 and so x LE 4.
For values of x=1, 2, 3, 4,
we can use (xy)(xz)=7x(x+y+z) to get (xy-7)(xz-7)=7x(x)+49.
For example
when x=1
we get (y-7)(z-7)=56,
so y=8, z=63; or y=9, z=35; or y=11, z=21; or y=14, z=15.
For x=2
we get y=4, z=42; or y=7, z=9.
For x=3
we get y=3, z=21; or y=5, z=7.
There are no new solutions for x=4,
so there are eight solutions in all.
[/SECRET]
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