Mathematical_Recreations-Kraitchik-2e

10~ Mathematical Recreations
Second: One of the sides of a primitive triangle is divisible
by 5. We shall prove this by showing that the product of
the three sides is divisible by 5. This product is
2ab(a 2 - b 2 ) (a 2 + b 2 ) = 2ab(a 4 - b 4 ).
if a or b is a multiple of 5, so is 2ab. If neither a nor b is a
multiple of 5, then a4 - b4 is. For a is then of one of the forms
5k ± 1, 5k ± 2. If we expand the fourth power of each of
these by the binomial theorem, we find at once that a 4 is of
the form 5h + 1. Similarly for b 4 •
Third: The area of a primitive right triangle is divisible
by 6, and the product of its sides by 60. For x is divisible by
4, x or y is divisible by 3, and x, y, or z is divisible by 5.
Hence the area, x!j, is divisible by 4~3, and xyz is divisible
by 4·3 ·5.
11. TRIGONOMETRIC RELATIONS. A primitive right triangle
will be called elementary or composite according as its
hypotenuse is prime or composite.
An angle is called arithmetical if all of its trigonometric
functions are rational. (The multiples of 90° are considered
to be arithmetical although some of their trigonometric
functions do not exist.) We shall show that an angle is arithmetical
if and only if it is an angle of a primitive right triangle,
or differs from such an angle by a multiple of 90°.
Let (x, y, z) be the sides of a primitive right triangle, and
let X, Y, Z be the angles opposite the respective sides. Z is
the right angle, and X and Yare the acute angles. Since all
the angles are determined by either acute angle we shall call
Y the angle of the right triangle. Then
. y a 2 - b2
SIn Y = - = = cos X,
z a + 2 b 2
x 2ab .
cos Y = - = = sm X
z a2 + b2 ,