Question

The sum of squares of three positive integers is 323. If the sum of squares of two numbers is twice the third, their product is

Answer 1

Let the three positive integers be x, y and z.

Sum of the squares of the three numbers =$x^2+y^2+z^2$

According to question,
Sum of the squares of the three numbers=323
$x^2+y^2+z^2=323$........(i)



The sum of the squares of the two numbers = $x^2+y^2$
According to question,
The sum of the squares of the two numbers =  Twice the third
The sum of the squares of the two numbers = 2z
$x^2+y^2=2z$......(ii)


Plug in  $x^2+y^2=2z$ from (ii) into (i):
$x^2+y^2+z^2=323$
$2z+z^2=323$
Subtract 323 from both sides:
$z^2+2z-323=0$
Solve the quadratic equation by factoring:
$z^2+19z-17z-323=0$
$z(z+19)-17(z+19)=0$
$(z+19)(x-17)=0$
Applying zero product rule:
z+19=0 or z-17=0
z=-19 or z=17

Since, the integers are positive, we reject z=-19.

Therefore, the 3rd number is z=17.


Plug in z=17 in (ii):
$x^2+y^2=2z$
$x^2+y^2=2(17)$
$x^2+y^2=34$
Express 34 as a sum of two squares, 34=9+25(only possibility):
$x^2+y^2=9+25$
$x^2+y^2=3^2+5^2$

Then, the other two numbers are x=3 and y=5.

Threfore, the product of the three numbers =x*y*z = 3*5*17=255


Answer: 255

Answer 2

Let us assume the three positive integers are x, y and z.

Given – The sum of  squares of  three numbers = 323

Therefore, x2 + y2 + z2 = 323 ------------1

Also given - The sum of squares of two numbers = Twice of the third interger

Therefore, x2 + y2 = 2z --------------2

Substitute the value of x2 + y2 = 2z from equation 2 in equation 1

2z + z2 = 323

z2 + 2z – 323 = 0

rearrange the term, we get

z2 + 19z – 17z - 323 = 0

z ( z + 19) – 17 ( z + 19) = 0

(z + 19) (z – 17) = 0

Z = -19 or z = 17

As given that all are positive integers, consider z = 17

Substitute the value of z in equation 2, we get

x2 + y2 = 2 * 17

x2 + y2 = 34

reassranging the number,

x2 + y2 =  9 + 25

x2 + y2 = 32 + 52

from above equation, we get

x = 5 and y = 3

Therefore, all the three positive integers are 3, 5 and 17

Answer - The product of all the three positive integer = 3 * 5 * 17 = 255.