Find all integers n for which the number (n+3) (n+1) is also an integer

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Solution

If $n = 0, (n + 3) (n - 1) = (0 + 3) (0 - 1) = 3 \times - 1 = - 3$ is an integer
If $n = 1, (n + 3) (n - 1) = (1 + 3) (1 - 1) = 4 \times 0 = 0$ is an integer.
If $n = - 1, (n + 3) (n - 1) = (- 1 + 3) (- 1 - 1) = 3 (- 2) = - 6$ is an integer
Similarly when n is replaced by an integer we find that $(n + 3) (n - 1)$ is also an integer
$∴$ $n = {. . . . . . . - 4, - 3, - 2, - 1, 0, 1, 2, 3, 4........}$

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Find all integers n for which the number (n + 3) (n − 1) is also an integer.

The sum of all positive integers n for which $\frac{1^{3} + 2^{3} . . . . + (2 n)^{3}}{1^{2} + 2^{2} + . . . n^{2}}$ is also an integer is.

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a_{n} = [\frac{n}{[\sqrt{n}]}]

[x]

x

x

n

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\frac{n^{3} + 3}{n^{2} + 7}

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