If , then 11^ m+2 + 12^ 2m-1 is divisible by: - Vidyadip

MCQ

If $\forall\text{m}\in\text{N},$ then $11^{m+2}+ 12^{2m-1}$ is divisible by:

A
$121$
B
$132$
✓
$133$
D
None of these

✓

Answer

Correct option: C.

$133$

To find the divisor of $11^{m+2}+ 12^{2m-1}$ by mathematic induction, the first step is to check for the smallest natural number,
i.e; for $m = 1.$
So, this reduces to $11^3+ 12^1$ or $11^4+ 1$.
So, the number when divided by $11$ leaves remainder $1.$
So, we can knock out options $A$ and $B$ as $121$ as well as $132$ are both divisible by $11$ and
hence their multiples will always be divisible by $11.$
Now, we have to check the divisibility of $11^{m+2}+ 12^{2m-1}$ by $133.$
For $m = 1, 11^4+ 1$ is not divisible by $133.$
So, we can knock out option $C.$

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