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prove that the even power of a natural number cannot be equal to $8k-3$

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Composition of linear and quadratic functions: find a value

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Q. prove that the even power of a natural number cannot be equal to

8k-3

Assume Even Power Property: Assume that an even power of a natural number can be written as $8k-3$ . Let $n$ be a natural number and $2m$ be its even power, where $m$ is a natural number. So, we have $n^{2m} = 8k - 3$ .
Consider Remainders: Consider the possible remainders when an even power of a natural number is divided by $8$ . An even power of a natural number is always a perfect square, and perfect squares can only have remainders of $0$ , $1$ , or $4$ when divided by $8$ .
Check Remainder of $8k-3$ : Now, let's check the remainder when $8k-3$ is divided by $8$ . $\newline$ $8k-3$ divided by $8$ gives a remainder of $-3$ , which is equivalent to a remainder of $5$ (since $-3 \mod 8 = 5$ ).
Evaluate Remainder Validity: We see that a remainder of $5$ is not possible for an even power of a natural number (as it can only be $0$ , $1$ , or $4$ ). $\newline$ Therefore, $n^{2m}$ cannot be equal to $8k-3$ .

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