Question

If ${9^7} - {7^9}$ is divisible by ${2^n}$ , then find the greatest value of $n$ ?
A) $4$
B) $6$
C) $10$
D) $8$

Answer 1

Hint:
Note that we have to find the highest power of the number two which divides the given number. So, it is very much important to express the given number in some power of the number two. Such a power of two will be the greatest value of the required unknown. We will find the maximum possible power which we are certain about.

Complete step by step solution:
The given expression is ${9^7} - {7^9}$.
Also, it is given that ${9^7} - {7^9}$ is divisible ${2^n}$ for some natural number $n$.
We will find the nearest number to both $9$ and $7$.
We know that ${2^3} = 8$ thus it is the nearest number as power of $2$.
We will express the given number as follows:
${9^7} - {7^9} = {\left( {8 + 1} \right)^7} - {\left( {8 - 1} \right)^9}$
From the second bracket we can take $ - 1$ common and it will change the sign as the power is odd.
${9^7} - {7^9} = {\left( {8 + 1} \right)^7} + 1{\left( {1 - 8} \right)^9}$
Now for any two numbers $a$ and $b$ the binomial expansion can be written as follows:
${\left( {a + b} \right)^n} = {}^n{C_0}{a^n}{b^0} + {}^n{C_1}{a^{n - 1}}{b^1} + \cdots + {}^n{C_n}{a^0}{b^n}$
Using this we can write the given expression as follows:
${9^7} - {7^9} = \left( {1 + {}^7{C_1}{8^1} + {}^7{C_2}{8^2} + \cdots + {}^7{C_7}{8^0}} \right) + 1{\left( {1 - {}^9{C_1}{8^1} + {}^9{C_2}{8^2} + \cdots + {}^9{C_9}{8^0}} \right)^9}$
Observe that the above relation can be expressed as:
${9^7} - {7^9} = 7 \times 8 + 9 \times 8 + {8^2}k$
Where $k$ is some nonzero constant such that for any $i$, we have ${2^i} \ne k$.
If we simplify the above relation then we can write:
${9^7} - {7^9} = {8^2}\left( {2 + k} \right)$
Now we will express the above term as power of $2$.
${9^7} - {7^9} = {2^6}\left( {2 + k} \right)$
Thus, the greatest value possible for $n$ is $6$.

Hence, the correct option is B.

Note:
Observe that the given numbers both exist in some neighbourhood of the asked number. Thus, we have to express the given difference as the power of the same number so that after using the binomial expansion we can express it as some power of the asked number.