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.
