Question

The remainder when \[{7^n} - 6n - 50(n \in N)\] is divided by 36 is,
A)22
B)23
C)1
D)19

Answer 1

Hint: Here we can make use of binomial theorem to expand the very first term of the sum given. After expansion, we will take out common powers of that number and place it with the remaining two terms.

Complete step by step answer:

Given that, \[{7^n} - 6n - 50(n \in N)\]
Let’s expand 7=6+1
\[{7^n} = {(6 + 1)^n}\]
Using binomial theorem,
\[\begin{gathered}
   \Rightarrow n{C_0}{6^n} + n{C_1}{6^{n - 1}} + n{C_2}{6^{n - 3}} + .....n{C_n}{6^0} \\
   \Rightarrow \left[ {n{C_0}{6^n} + n{C_1}{6^{n - 1}} + n{C_2}{6^{n - 3}} + .....n{C_{n - 2}}{6^2}} \right] + 6n + 1 \\
   \Rightarrow \left[ {36N} \right] + 6n + 1 \\
\end{gathered} \]
36N is the number because it is common for all powers of 6 in bracket.
This is expansion of the term \[{7^n}\]
Now again given that,
\[{7^n} - 6n - 50\]=
\[\begin{gathered}
   \Rightarrow \left[ {36N} \right] + 6n + 1 - 6n - 50 \\
   \Rightarrow 36N - 49 \\
   \Rightarrow 36N - 72 + 23 \\
\end{gathered} \]
Here the term \[36N - 72\]is divisible by 36 totally, but the number 23 is not divisible by 36.
So the remainder will be 23.
Option B is the correct answer.

Note: Here student can expand the series as 7=8-1 but remember The number should be divisible by 36 and powers of 8 are not of that kind. So 7=6+1 is the correct way.

Additional information:
If n is any positive integer, then
           \[\begin{gathered}
  {(x + a)^n} = n{C_0}{x^n} + n{C_1}{x^{n - 1}}a + n{C_2}{x^{n - 2}}{a^2} + .......n{C_n}{a^n} \\
  {(x + a)^n} = \sum\nolimits_{r = 0}^n {n{C_r}{x^{n - r}}{a^r}} \\
\end{gathered} \]
           This is called the binomial theorem.
Here \[n{C_0}\],\[n{C_1}\],\[n{C_2}\]…….\[n{C_n}\] are binomial coefficients.
Where \[n{C_r} = \dfrac{{n!}}{{r!\left( {n - r} \right)!}}\] \[0 < r < n\].
Total number of terms in the expansion of \[{(x + a)^n}\] is n+1.
The sum of the indices of x and a in each term is n.
Applications of the binomial theorem:
The binomial theorem is used in economic prediction. This helps economists to predict that the economy will fall or bounce.
It is also used in architecture or civil engineering to predict the estimates of cost and time required for that project.
Weather forecasting is another field in which the binomial theorem is used.