They look like coefficients of cubic polynomial!

Number Theory Level 3

If x , y x,y and z z are positive integers satisfying x y z + x y + y z + x z + x + y + z = 384 xyz+xy+yz+xz+x+y+z = 384 , find the value of x + y + z x+y+z .

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Puneet Pinku by Puneet Pinku , 20, India

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1 solution

x y z + x y + y z + x z + x + y + z = 384 xyz + xy + yz + xz + x + y + z = 384

( x + 1 ) ( y + 1 ) ( z + 1 ) = x y z + x y + y z + x z + x + y + z + 1 (x + 1) (y + 1) (z + 1) = xyz + xy + yz + xz + x + y + z + 1

( x + 1 ) ( y + 1 ) ( z + 1 ) = 384 + 1 = 385 (x + 1) (y + 1) (z + 1) = 384 + 1 = 385

( x + 1 ) ( y + 1 ) ( z + 1 ) = ( 5 ) ( 7 ) ( 11 ) (x + 1) (y + 1) (z + 1) = (5) (7) (11)

( x + 1 ) + ( y + 1 ) + ( z + 1 ) = 5 + 7 + 11 (x + 1) + (y + 1) + (z + 1) = 5 + 7 + 11

x + y + z + 3 = 23 x + y + z + 3 = 23

x + y + z = 20 x + y + z = 20

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