# Q. 222^333 + 333^222 is divisible by which of the following numbers?

a) 2 and 3 but not 37

b) 3 and 37 but not 2

c) 2 and 37 but not 3

d) 2, 3 and 37

Correct answer: b) 3 and 37 but not 2

#### Question from UPSC Prelims 2024 CSAT

**Explanation : **

## 222^333 + 333^222

1) Let’s start with the expression: 222^333 + 333^222

2) Factor out the common factors:

222 = 2 × 3 × 37

333 = 3 × 111

3) Now we can rewrite our expression:

(2 × 3 × 37)^333 + (3 × 3 × 37)^222

4) Let’s focus on the divisibility by each number:

For 2:

– The first term (2 × 3 × 37)^333 is clearly even (divisible by 2).

– The second term (3 × 111)^222 is odd (not divisible by 2).

– Even + Odd = Odd

– Therefore, the sum is not divisible by 2.

For 3:

– Both terms contain 3 as a factor.

– Therefore, the sum is divisible by 3.

For 37:

– Both terms contain 37 as a factor.

– Therefore, the sum is divisible by 37.

Therefore, the correct answer remains b) 3 and 37 but not 2.