Question

If ${9^7} - {7^9}$ is divisible by 2n, then find the greatest value of n ?

Answer 1

Hint :We will solve this question using binomial expansion.
Binomial expansion – The binomial theorem is the method of expanding an expression which has been raised to any finite power.
Binomial expression – A binomial expression is an algebraic expression which contains two dissimilar terms. Ex. $a + b,{a^3} + {b^3}$ etc.
Binomial theorem :
${(a + b)^n}{ = ^n}{C_0}{a^n}{b^0}{ + ^n}{C_1}{a^{(n - 1)}}{b^1}{ + ^n}{C_2}{a^{(n - 2)}}{b^2} + .....{ + ^n}{C_n}{a^0}{b^n}$


Complete step by step solution :
${9^7} - {7^9}$
We can write it as, ${(8 + 1)^7} + {(8 - 1)^9}$
or, ${(1 + 8)^7} + ( - 1) \times {(1 - 8)^9}$
${(1 + 8)^7} - {(1 - 8)^9}$
By using binomial expansion :
\[{(1 + x)^n}{ = ^n}{C_0}{(1)^n}{x^0}{ + ^n}{C_1}{(1)^{(n - 1)}}{x^1} + .....{ + ^n}{C_n}{(1)^0}{x^n}\]
${(1 + 8)^7} - {(1 - 8)^9}$
$ = \left[ {\left( {^7{C_0}{8^0}{ + ^7}{C_1}{8^1}{ + ^7}{C_2}{8^2}{ + ^7}{C_3}{8^3} + .....{ + ^7}{C_7}{8^7}} \right)} \right]$$ - \left[ {\left( {^9{C_0}{{( - 8)}^0}{ + ^9}{C_1}{{( - 8)}^1}{ + ^9}{C_2}{{( - 8)}^2}{ + ^9}{C_3}{{( - 8)}^3} + .....{ + ^9}{C_9}{{( - 8)}^9}} \right)} \right]$
$ = 1{ + ^7}{C_1}{8^1}{ + ^7}{C_2}{8^2}{ + ^7}{C_3}{8^3}{ + ^7}{C_4}{8^4}{ + ^7}{C_5}{8^5}{ + ^7}{C_6}{8^6}{ + ^7}{C_7}{8^7}$
$ - 1{ + ^9}{C_1}{8^1}{ - ^9}{C_2}{8^2}{ + ^9}{C_3}{8^3}{ - ^9}{C_4}{8^4}{ + ^9}{C_5}{8^5}{ - ^9}{C_6}{8^6}{ + ^9}{C_7}{8^7}{ - ^9}{C_8}{8^8}{ + ^9}{C_9}{8^9}$
$ = (7 \times 8) + (9 \times 8) + {8^2}\left[ {\left( {^7{C_2}{ + ^7}{C_3}{8^1}{ + ^7}{C_4}{8^2} + .....} \right) - \left( {^9{C_2}{ - ^9}{C_3}{8^1} + .....} \right)} \right]$
Term : $\left[ {\left( {^7{C_2}{ + ^7}{C_3}{8^1}{ + ^7}{C_4}{8^2} + .....} \right) - \left( {^9{C_2}{ - ^9}{C_3}{8^1} + .....} \right)} \right]$ is constant.
So, let this term be k
Then, we have
$ = (7 \times 8) + (9 \times 8) + {8^2}k$
$ = 56 + 72 + {8^2}k$
$ = 132 + {8^2}k$
$ = {8^2}(4 + k)$
So,
${9^7} - {7^9} = {8^2}(4 + k)$
$ = 2 \times 32(4 + k)$
As, we have to prove ${9^7} - {7^9}$ as a multiple of 2.
So, we write ${8^2} = 64$ as $2 \times 32$.
And also, $32(4 + k)$ is another constant.
Therefore, ${9^7} - {7^9} = 2(32c) = 2n$
Where $n = 32$

Note : Make sure you do not apply the formula/concept of permutation here .There is a high chance of students getting confused whether to apply permutation or combination in these kinds of problems.