JAMIA 2023 — Mathematics PYQ

Step 1: Test for $n=1$

Since the expression must be divisible by $64$ for all natural numbers, it must specifically be divisible for $n=1$.

Substituting $n=1$:

For this to be divisible by $64$:

The closest multiple of $64$ to $65$ is $64$ itself.

Step 2: Verify using Binomial Expansion

To ensure this works for all $n$, let's rewrite $49^n$ as $(1+48{)}^n$:

Since $48^2=2304$, and $2304$ is divisible by $64$ ($2304=64\times 36$), all terms from the third term onwards are divisible by $64$.

Thus:

Now, substitute this back into the original expression:

For the expression to be divisible by $64$:

Step 3: Find the least negative value

The values of $\lambda$ satisfying this are $-1,-65,-129,\dots$

The least negative value (the negative value with the smallest magnitude) is:

Final Answer:

The least negative value of $\lambda$ is $-1$.