Divisibility Problems
Category : JEE Main & Advanced
To show that an expression is divisible by an integer
(i) If \[a,p,n,r\] are positive integers, then first of all we write \[{{a}^{pn+r}}={{a}^{pn}}.\,{{a}^{r}}={{({{a}^{p}})}^{n}}.\,{{a}^{r}}.\]
(ii) If we have to show that the given expression is divisible by \[c\].
Then express, \[{{a}^{p}}=[1+({{a}^{p}}-1]\], if some power of \[({{a}^{p}}-1)\] has c as a factor. \[{{a}^{p}}=[2+({{a}^{p}}-2)]\], if some power of \[({{a}^{p}}-2)\] has c as a factor.
\[{{a}^{p}}=[k+({{a}^{p}}-k)],\,\]if some power of \[({{a}^{p}}-k)\] has c as a factor.
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