The difference between an integer n and its c | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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?

Similar Questions

If $n$ is an integer with $0 \leq n \leq 11$,then the minimum value of $n!(11-n)!$ is attained when a value of $n$ equals to

If $P(n, r) = 1680$ and $C(n, r) = 70$,then $69n + r! = \dots$.

The number of $5$-digit natural numbers such that the product of their digits is $36$ is:

Consider the following statements:
$I$. The number of positive integral solutions of $x_1+x_2+x_3+x_4=10$ is $286$.
$II$. If $25! = 10^n \times k, (k \in N)$, then $n=6$.
Which one of the following options is true?

In a circus,there are $10$ cages for accommodating $10$ animals. Out of these,$4$ cages are so small that $5$ out of the $10$ animals cannot enter them. In how many ways is it possible to accommodate the $10$ animals in these $10$ cages?

Vedclass Test Series

Exam Paper Generator

Online Exam Module