CSAT 2024

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.

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