Question

Which of the following statements is/are true?
I. \(4^{10} + 6^{10}\) is divisible by 52.
II. \(7^{15} + 64^5\) is divisible by 11.
III. \(2^{20} - 49^{10}\) is divisible by 9.
IV. \(3^{15} - 8^5\) is divisible by 5.

• A. Only I and II
• B. I, II, III and IV
• C. Only I
• D. Only I, II and III

Hint:

Use modular arithmetic to quickly check divisibility rules. \(A \equiv B \pmod n \implies A - B\) is divisible by \(n\).

Answer 1

The Correct Option is D

Statement I: \(52 = 4 \times 13\). Term is divisible by 4. Mod 13: \(4^{10} + 6^{10} \equiv 9 + (-9) \equiv 0\). (True) Statement II: Mod 11. \(7^{15} + 64^5 \equiv (7^3)^5 + 9^5 \equiv 343^5 + (-2)^5 \equiv 2^5 - 32 \equiv 32 - 32 \equiv 0\). (True) (Note: \(343 \equiv 2 \pmod{11}\)). Statement III: Mod 9. \(2^{20} - 49^{10} \equiv 2^{20} - 4^{10} \equiv 2^{20} - (2^2)^{10} \equiv 0\). (True) Statement IV: Mod 5. \(3^{15} - 8^5 \equiv 3^{15} - 3^5 \equiv 3^5(3^{10} - 1)\). \(3^{10} \equiv (3^2)^5 \equiv (-1)^5 \equiv -1\). Value \(\equiv 3^5(-1-1) \equiv -2 \cdot 3^5 \not\equiv 0\). (False)