JEE Mathematics 2023 Question with Solution

Given:

• Statement (S1): $2023^{2022} - 1999^{2022}$
• Statement (S2): $13(13^{n} - 1) - 13$

Find: Which of the two statements is correct.

For (S1), the solution states that the expression is not divisible by $8$, so statement (S1) is false.

For (S2), use the binomial expansion of $13 = 1+12$:

for some integer $K$.

Hence the expression in (S2) is divisible by $144$, so statement (S2) is true for infinitely many $n \in \mathbb{N}$.

Therefore, only statement (S2) is correct, so the correct option is B.

Given:

• $2023^{2022} - 1999^{2022}$
• $13(13^{n} - 1) - 13$

Find: Determine which statement is true.

Check (S1) modulo $8$. Since $2023 \equiv 7 \equiv -1 \pmod{8}$ and $1999 \equiv 7 \equiv -1 \pmod{8}$, and the exponent $2022$ is even,

and

Therefore,

So (S1) is true.

Now check (S2):

Using $13 = 1+12$,

The provided solution’s working groups all remaining terms into a multiple of $144$, showing divisibility by $144$ for infinitely many $n$.

Thus the extracted page concludes that only (S2) is correct, and the correct option is B.