Question

Among the statements:

• A. only (S₁) is correct
• B. both (S₁) and (S₂) are correct
• C. only (S₂) is correct
• D. both (S₁) and (S₂) are incorrect

Answer 1

The Correct Option is B

The correct option is (B): Only statement-1 is true
∵ (2002)²⁰²³ = 8 m
∵ (2002)²⁰²³ is divisible by 8 and (1919)²⁰⁰² is not divisible by 8
∴ (2002)²⁰²³ - (1919)²⁰⁰² is not divisible by 8.
Also, 13.(13)ⁿ - 12n - 13
= 13(1+12)ⁿ - 12n - 13
= 13 [1+12n + ⁿC₂ 12² + --] - 12n - 13
= 144n + 144ⁿC₂ + --
= 144 [n + ⁿC₂ + --]
= 144K
∴, the option (B) is the correct option.

Answer 2

For (S1), we can apply modulo operations to prove that \(2023^{2022} - 1999^{2022}\) is divisible by 8. For (S2), using the given formula, we can show the divisibility of \(13(13^n - 1) - 13\) by 144 for infinitely many \(n\). Thus, both statements are true.