To the squares!

Let x x , y y and z z be all positive integers with x y = 21 xy=21 , x z = 28 xz = 28 and y z = 12 yz=12 . Find x 2 + y 2 + z 2 x^2 + y^2 + z^2 .

74 62 39 50 28

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3 solutions

x y x z = 21 28 x 2 y z = 21 28 12 x 2 = 21 28 x 2 = 21 28 12 = 7 2 x = 7 y = 21 x = 3 z = 28 x = 4 \begin{aligned} xy \cdot xz & = 21 \cdot 28 \\ x^2 {\color{#3D99F6}yz} & = 21 \cdot 28 \\ {\color{#3D99F6}12} x^2 & = 21 \cdot 28 \\ x^2 & = \frac {21 \cdot 28}{12} = 7^2 \\ \implies x & = 7 \\ \implies y & = \frac {21}x = 3 \\ \implies z & = \frac {28}x = 4 \end{aligned}

x 2 + y 2 + z 2 = 49 + 9 + 16 = 74 \implies x^2 + y^2 + z^2 = 49 + 9 + 16 = \boxed{74}

The answer should be 74, not 33.

Jessey Uche-Nwichi - 4 years, 4 months ago

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Thanks. What had happened to me. Thirty-three must be the answer of previous solution I wrote.

Chew-Seong Cheong - 4 years, 4 months ago

Take the product of the equations to get ( x y ) ( x z ) ( y z ) = 21 28 12 = 3 2 4 2 7 2 xy)(xz)(yz)=21*28*12=3^2*4^2*7^2 , or x y z = 3 4 7 xyz=3*4*7 . x y z x y \frac{xyz}{xy} = 3 4 7 21 \frac{3*4*7}{21} , z = 4. z=4. Similarly, x = 7 x=7 and y = 3. x 2 + y 2 + z 2 = 3 2 + 4 2 + 7 2 = 5 2 + 7 2 = 74 y = 3. x^2 + y^2 + z^2=3^2 + 4^2 + 7^2= 5^2+7^2=74

Thomas Horstkamp
Feb 23, 2017

x y x z \frac{xy}{xz} = y z \frac{y}{z} = 3 4 \frac{3}{4}

y = y = 3 4 \frac{3}{4} z z

x y y z \frac{xy}{yz} = x z \frac{x}{z} = 7 4 \frac{7}{4}

x = x = 7 4 \frac{7}{4} z z

x + y + z = x+y+z = 7 4 \frac{7}{4} z + z + 3 4 \frac{3}{4} z + z + z = z = 7 2 \frac{7}{2} z z

x z × y z = x y z 2 = xz \times yz = xyz^2 = 336 336

x y z 2 x y \frac{xyz^2}{xy} = z 2 = z^2 = = 336 21 \frac{336}{21} = 16 =16

( x + y + z ) 2 (x+y+z)^2 = x 2 + y 2 + z 2 + 2 x y + 2 x z + 2 y z = = x^2 + y^2 +z^2 + 2xy +2xz +2yz = ( 7 z 2 (\frac{7z}{2} ) 2 )^2 = 196 =196

x 2 + y 2 + z 2 + 2 x y + 2 x z + 2 y z = 196 x^2+y^2+z^2+2xy+2xz+2yz = 196

x 2 + y 2 + z 2 + 2 ( 21 ) + 2 ( 28 ) + 2 ( 12 ) = 196 x^2 + y^2 +z^2 +2(21)+2(28)+2(12) = 196

x 2 + y 2 + z 2 + 42 + 56 + 24 = 196 x^2 +y^2 +z^2 + 42 + 56 + 24 = 196

x 2 + y 2 + z 2 = 74 x^2 +y^2 +z^2 = 74

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