Let $XYZ$ be a three-digit number, where $(X + Y + Z)$ is not a multiple of 3. Then $(XYZ + YZX + ZXY)$ is…
Let $XYZ$ be a three-digit number, where $(X + Y + Z)$ is not a multiple of 3. Then $(XYZ + YZX + ZXY)$ is not divisible by
3
9
37
$(X + Y + Z)$
Solution
$XYZ + YZX + ZXY = 111(X+Y+Z) = 3 \times 37 \times (X+Y+Z)$. So the sum is always divisible by 3, by 37, and by $(X+Y+Z)$. It is divisible by 9 only if $(X+Y+Z)$ is a multiple of 3. Since $(X+Y+Z)$ is given as not a multiple of 3, the sum is not divisible by 9.