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 ✓ UPSC's answer
- (c) 2 and 37 but not 3
- (d) 2, 3 and 37
Why the answer is (b)
• Check divisibility by 2: 222 is even, so 222^333 is even. 333 is odd, so 333^222 is odd. The sum of an even and an odd number is odd, hence not divisible by 2.
• Check divisibility by 3: 222 is divisible by 3 (sum of digits 6), so 222^333 is divisible by 3. 333 is divisible by 3 (sum of digits 9), so 333^222 is divisible by 3. The sum is divisible by 3.
• Check divisibility by 37: Note that 222 = 6 * 37, so 222^333 is divisible by 37. Also, 333 = 9 * 37, so 333^222 is divisible by 37. The sum is divisible by 37.
• Since the expression is divisible by 3 and 37 but not by 2, option (b) is correct.
Why the other options are wrong
- (a) 2 and 3 but not 37
- The expression is not divisible by 2 because it is the sum of an even and an odd number.
- (c) 2 and 37 but not 3
- The expression is not divisible by 2, so any option claiming divisibility by 2 is incorrect.
- (d) 2, 3 and 37
- The expression is not divisible by 2, so it cannot be divisible by 2, 3, and 37 simultaneously.
Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2024, held on 16 June 2024. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.