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.
