Question

The statement $P\left( n \right)={{9}^{n}}-{{8}^{n}}$ when divided by 8 always leaves the remainder.
[a] 2
[b] 3
[c] 1
[d] 7

Answer 1

Hint: Write 9 as 8+1 and use the fact that the expansion of ${{\left( x+y \right)}^{n}}$ is given by ${{\left( x+y \right)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}}$

Complete step-by-step answer:
Hence write the expansion of ${{\left( 1+8 \right)}^{n}}$.
Subtract ${{8}^{n}}$ from both sides of the expansion and hence find the remainder obtained on dividing ${{9}^{n}}-{{8}^{n}}$ by 8.
Alternatively, put 8 = x and hence write P(n) in terms of x, i.e. $g\left( x \right)=P\left( n \right)={{\left( x+1 \right)}^{n}}-{{x}^{n}}$. Use remainder theorem, which states that the remainder obtained on dividing p(x) by x-a is given by $p\left( a \right)$. Hence find the remainder obtained on dividing g(x) by x and hence find the remainder obtained on dividing P(n) by 8.

We have $P\left( n \right)={{9}^{n}}-{{8}^{n}}$
We know that 9 = 1+8
Hence, we have
$P\left( n \right)={{\left( 1+8 \right)}^{n}}-{{8}^{n}}$
We know from the binomial theorem that the expansion of ${{\left( x+y \right)}^{n}}$ is given by ${{\left( x+y \right)}^{n}}=\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{x}^{n-r}}{{y}^{r}}}$
Hence, we have
$P\left( n \right)=1{{+}^{n}}{{C}_{1}}8{{+}^{n}}{{C}_{2}}{{8}^{2}}+\cdots {{+}^{n}}{{C}_{n-1}}{{8}^{n-1}}{{+}^{n}}{{C}_{n}}{{8}^{n}}-{{8}^{n}}$
Hence, we have
$P\left( n \right)=8\left( ^{n}{{C}_{1}}{{+}^{n}}{{C}_{2}}8+\cdots {{+}^{n}}{{C}_{n-1}}{{8}^{n-2}} \right)+1=8k+1,k\in \mathbb{N}$
Hence, by Euclid's division lemma, the remainder obtained on dividing P(n) by 8 is 1.
Hence option [c] is correct.

Note: Alternative Solution:
Let x = 8
Hence, we have $P\left( n \right)=g\left( x \right)={{\left( x+1 \right)}^{n}}-{{x}^{n}}$
We know from the remainder theorem that the remainder obtained on dividing p(x) by x-a is given by $p\left( a \right)$
Hence, the remainder obtained on dividing g(x) by x is given by g(0)
Now, we have
$g\left( 0 \right)={{\left( 0+1 \right)}^{n}}-{{0}^{n}}=1$
Hence the remainder obtained on dividing P(n) by 8 is 1.
Hence option [c] is correct.