Correct Answer - Option 2 : 9 

Given:

XYZ is a three-digit number

Where (X + Y + Z) is not a multiple of 3

Concept used:

General form of a number XYZ = X × 100 + Y × 10 + Z

Calculation:

The general form of XYZ = 100X + 10Y + Z.

The general form of YZX = 100Y + 10Z + X

The general form of ZXY = 100Z + 10X + Y.

Using above, add the three numbers, we get ;

⇒ XYZ + YZX + ZXY = (100X + 10Y + Z) + (100Y + 10Z + X) + (100Z + 10X + Y)

⇒ XYZ + YZX + ZXY = 111X + 111Y + 111Z

⇒ XYZ + YZX + ZXY = 111(X + Y + Z).

Now, it is clear that 111 is the common factor.

So, XYZ + YZX + ZXY is divisible by 111 and it's factors

Factors of 111 are 3 and 37. That eliminate option 3 and 37

But it is given that (X + Y+ Z) is not multiple of 3

⇒ (XYZ + YZX + ZXY) is not divisible by 9.

⇒ (XYZ + YZX + ZXY) is not divisible by 9.