n is a positive integer. Show that, for all n, $(5n+3)^2-(5n-3)^2$ is a multiple of 4.

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Worked solution

1. Recognise and apply the difference of two squares.

$(5n+3)^2-(5n-3)^2$
$=[(5n+3)+(5n-3)][(5n+3)-(5n-3)]$

2. Simplify and express the result as 4 times an integer.

$=[10n][6]$
$=60n$
$=4(15n)$
Since $n$ is a positive integer, $15n$ is an integer. Hence $(5n+3)^2-(5n-3)^2$ is a multiple of $4$ .

Answer

$(5n+3)^2-(5n-3)^2=60n=4(15n)$ . Since $15n$ is an integer, $(5n+3)^2-(5n-3)^2$ is a multiple of 4.

AI analysis — may contain mistakes.

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