From the ten books that you've recently bought but not read, you plan to take four with you on vacation. How many different sets
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Let 
be an integer. Suppose there is a triangle with legs of length 16 and 
. Then by the Pythagorean theorem, the length of the hypotenuse should be
The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if
and 
are integers, then so are 
, 
, and 
.
However,
is not a perfect square trinomial, which means for any integer , the length of the hypotenuse is not an integer, so such a triangle doesn't exist.
The formulas for Pythagorean triples say that if the legs are integers, then so must be the hypotenuse, because if
However,