Question

Чи при всіх значеннях n значення виразу (n + 29)(n+3)-(n+7)(n+1) кратне 8? помогите прошу!!!

Answer 1

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Answer 2

Checking the Divisibility of an Expression by 8

To determine whether the expression \((n + 29)(n + 3) - (n + 7)(n + 1)\) is divisible by 8 for all values of \(n\), we can expand the expression and then analyze its divisibility by 8.

Expansion of the Expression

Expanding the given expression, we get:

\((n + 29)(n + 3) - (n + 7)(n + 1)\)

Expanding further:

\(n^2 + 3n + 29n + 87 - (n^2 + 1n + 7n + 7)\)

Simplifying:

\(n^2 + 32n + 87 - n^2 - 8n - 7\)

Combining like terms:

\(24n + 80\)

Analyzing Divisibility by 8

To check if the expression is always divisible by 8, we need to verify if \(24n + 80\) is always a multiple of 8 for all values of \(n\).

Claim: The expression \(24n + 80\) is always divisible by 8 for all values of \(n\).

This claim is supported by the fact that any multiple of 24 is divisible by 8, and adding 80 to a multiple of 24 will still result in a number that is divisible by 8.

Therefore, the expression \((n + 29)(n + 3) - (n + 7)(n + 1)\) is indeed always divisible by 8 for all values of \(n\) .

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