Question

which of the numbers 4370, 1885, 3185 can be a hypothenuse in a primitive pythagoreantriangle? explain. when possible find such a triple. (note that we are looking to represent43702 as a sum of squares, not 4370. recall also what is required of numbers to be a primitivetriple)

Answer 1

None of the numbers 4370, 1885, 3185 can be a hypotenuse in a primitive Pythagorean triangle.

Step-by-step explanation:

In a primitive Pythagorean triple, all three numbers are positive integers that are relatively prime to each other. To determine if a number can be the hypotenuse in a primitive Pythagorean triangle, we need to check if it can be expressed as the sum of two squares. Let's check each number:

For 4370:

4370 = 2^1 * 5 * 19 * 23

Since 4370 has prime factors larger than 2, it cannot be a sum of two squares and therefore cannot be a hypotenuse in a primitive Pythagorean triangle.

For 1885:

1885 = 5 * 13 * 29

Since 1885 has prime factors larger than 2, it cannot be a sum of two squares and therefore cannot be a hypotenuse in a primitive Pythagorean triangle.

For 3185:

3185 = 5 * 11 * 59

Since 3185 has prime factors larger than 2, it cannot be a sum of two squares and therefore cannot be a hypotenuse in a primitive Pythagorean triangle.

None of the given numbers can be a hypotenuse in a primitive Pythagorean triangle.