The difference between an integer n and its c | vedclass

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(B) Let the integer be $n$. We want to examine the expression $n^3 - n$.Factorizing the expression,we get $n^3 - n = n(n^2 - 1) = n(n - 1)(n + 1)$.Thi

Vedclass · Accepted Answer

$6$

Vedclass · Answer

(B) Let the integer be $n$. We want to examine the expression $n^3 - n$.Factorizing the expression,we get $n^3 - n = n(n^2 - 1) = n(n - 1)(n + 1)$.This is the product of three consecutive integers: $(n - 1)$,$n$,and $(n + 1)$.In any set of three consecutive integers,at least one must be even (divisible by $2$) and exactly one must be divisible by $3$.Since the product is divisible by both $2$ and $3$,and $	ext{gcd}(2, 3) = 1$,the product must be divisible by $2 	imes 3 = 6$.Thus,$n^3 - n$ is always divisible by $6$.

The difference between an integer $n$ and its cube $n^3$ is always divisible by which of the following?

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