15. 222^333 + 333^222 is divisible by which of the following numbers ?

1. A. 2 and 3 but not 37
2. B. 3 and 37 but not 2
3. C. 2 and 37 but not 3
4. D. 2, 3 and 37

Answer: B

Explanation

Let’s analyze the divisibility of the expression 222^333 + 333^222 by 2, 3, and 37.
1. Divisibility by 2: 222^333 is an even number (even base raised to any positive integer power is even). 333^222 is an odd number (odd base raised to any positive integer power is odd). The sum of an even and an odd number is always odd. Therefore, the expression is not divisible by 2.
2. Divisibility by 3: 222 is divisible by 3 (sum of digits 2+2+2=6). So, 222^333 is divisible by 3. 333 is divisible by 3 (sum of digits 3+3+3=9). So, 333^222 is divisible by 3. The sum of two numbers divisible by 3 is also divisible by 3. Thus, the expression is divisible by 3.
3. Divisibility by 37: 222 = 6 × 37. So, 222^333 is divisible by 37. 333 = 9 × 37. So, 333^222 is divisible by 37. The sum of two numbers divisible by 37 is also divisible by 37. Thus, the expression is divisible by 37.
Based on this analysis, the expression is divisible by 3 and 37, but not by 2. This question tests number properties and divisibility rules.