CSAT 2024 Solutions - 222^333 + 333^222 is divisible by which of the following numbers?

222³³³ + 333²²² is divisible by which of the following numbers?

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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

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Answer: B. 3 and 37 but not 2

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Explanation:

Step 1: Check divisibility by 2

• 222 is even, so 222³³³ is even.
• 333 is odd, so 333²²² is odd.
• The sum of an even number (222³³³) and an odd number (333²²²) is odd.

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• Thus, 222³³³ + 333²²² is not divisible by 2.

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Step 2: Check divisibility by 3

• Using the property of divisibility by 3, a number is divisible by 3 if the sum of its digits is divisible by 3.

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• Sum of digits of 222 = 2 + 2 + 2 = 6, so 222 is divisible by 3. Therefore, 222³³³ is divisible by 3.
• Sum of digits of 333 = 3 + 3 + 3 = 9, so 333 is divisible by 3. Therefore, 333²²² is divisible by 3.

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• Since both terms 222³³³ and 333²²² are divisible by 3, their sum is also divisible by 3.

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• Thus, 222³³³ + 333²²² is divisible by 3.

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Step 3: Check divisibility by 37

• To check divisibility by 37, use modular arithmetic:

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• 222 mod 37 = 222 - 37 × 6 = 222 - 222 = 0.
• Hence, 222 ≡ 0 mod 37, and 222³³³ ≡ 0 mod 37.

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• 333 mod 37 = 333 - 37 × 9 = 333 - 333 = 0.
• Hence, 333 ≡ 0 mod 37, and 333²²² ≡ 0 mod 37.

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• Since both terms are divisible by 37, their sum is also divisible by 37.

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• Thus, 222³³³ + 333²²² is divisible by 37.

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Final Answer:

The expression 222³³³ + 333²²² is divisible by 3 and 37 but not 2.

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