x, y, z are integer that are side of an obtuse-angled triangle. If xy = 4,…

We are given integer sides x, y, z of a triangle with xy = 4. Find integer z so the triangle is obtuse.

The integer factor pairs (x, y) with product 4 are (1, 4), (2, 2), and (4, 1). These lead to two distinct side-sets to check: {1, 4, z} and {2, 2, z}.

• Case {2, 2, z}: Triangle inequality requires z < 2 + 2 = 4, so integer possibilities are z = 1, 2, 3.

  Check for obtuse: a triangle is obtuse when the square of the largest side is greater than the sum of the squares of the other two sides.

  For z = 1 or 2 the triangle is acute. For z = 3, the largest side is 3 and 3^2 = 9 while 2^2 + 2^2 = 8, so 9 > 8 and the triangle 2, 2, 3 is obtuse.

• Case {1, 4, z}: Triangle inequality requires z < 1 + 4 = 5 and also z > 4 - 1 = 3, so the only integer possibility is z = 4.

  For sides 1, 4, 4 the largest side is 4 and 4^2 = 16 while the sum of the squares of the other two is 4^2 + 1^2 = 17, so 16 < 17 and the triangle is acute.

Therefore, the only integer z that produces an obtuse triangle given xy = 4 is 3.

Answer: 3