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If n is a natural number, then find the numbers by which $9^{2 n} - 4^{2 n}$ is divisible.

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None of these

both 13 and 5

answer is C.

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Detailed Solution

Given:

9^{2 n} - 4^{2 n}

where n is a natural number.
Use the following theorem for solving theorem:

a + b

and

a - b

are always factors of

a^{2 n} - n^{2 n}

.
In other words,

a^{2 n} - n^{2 n}

is always divisible by

a + b

and

a - b

provided n is a natural number.
Assume the values of a and b as 9 and 4 respectively. Then,

a^{2 n} - n^{2 n} = 9^{2 n} - 4^{2 n}

\Rightarrow 9^{2 n} - 4^{2 n}

is always divisible by

9 + 4 and 9 - 4

\Rightarrow 9^{2 n} - 4^{2 n}

is always divisible by 13 and 5.
Therefore, if n is a natural number, then

9^{2 n} - 4^{2 n}

is divisible by 13 and 5.
Hence, option 3 is correct.

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