# Q. 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

(a) 3

(b) 9

(c) 37

(d) (X + Y + Z)

Correct Answer: (b) 9

#### Question from UPSC Prelims 2020 CSAT Paper

**Model Answer:**

## XYZ be a three-digit number

Let’s write XYZ in expanded form, where X, Y, and Z are the digits in the hundreds, tens, and units places, respectively.

XYZ = 100X + 10Y + Z

Similarly, we can write YZX and ZXY in expanded form:

YZX = 100Y + 10Z + X

ZXY = 100Z + 10X + Y

Adding these three expressions, we get:

XYZ + YZX + ZXY = 100X + 10Y + Z + 100Y + 10Z + X + 100Z + 10X + Y

= 111X + 111Y + 111Z=111(X + Y + Z)

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