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Positive integers x, y, z satisfy x y+z=160 . Compute the smallest possible value of x+y z .

Correct answer

50

Step-by-step solution

aligned & x+y z= 160-z y +y z & 160 y + z (y^2-1 ) y = (160-z) y + z y^2 y = 160-z y +z y aligned At particle value of z it is greater than equal to 2 z(160-z) least value is 2 z(160-z) but integer also Now least for least value z is also Case I z=1, x+y z= 159 y +y minimum value is at y=3 which is 56 Case II z=2, x+y z= 158 y +2 y minimum value at y=2 which is 83 (rejected) Case III z=3, x+y z= 157 y +3 y minimum at y=1 which is 160 (rejected) Case IV z=4, x+y z= 156 y +4 y minimum at y=6 which is 50 (accept) Case

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