Question

What is remainder when \[{7^{84}}\] is divided by $2402?$
(A) $1$
(B) $6$
(C) $2401$
(D) None of these

Answer 1

Hint: The Binomial theorem tells us how to expand expressions of the form ${(a + b)^n}$ . The larger the power is, the harder it is to expand expressions directly. But with the Binomial theorem, the process is relatively easy and fast. First we simplify the given function and then calculate. After simplifying we use the Binomial theorem and get the remainder term.
Formula of expand by using binomial theorem, we get \[{(a + b)^n}{ = ^n}{C_0}{(a)^0} \times {(b)^n}{ + ^{n - 1}}{C_1}{(a)^1} \times {(b)^{n - 1}}{ + ^{n - 2}}{C_2}{(a)^2} \times {(b)^{n - 2}} + .....{ + ^n}{C_n}{(a)^n} \times {(b)^0}\]

Complete step-by-step solution:
Given the function ${7^{84}}$
Simplifying the above function and we get
$ = {({7^4})^{21}}$
We know that the ${7^4} = 7 \times 7 \times 7 \times 7$
$ = 49 \times 49$
$ = 2401$
Put this in above function and we get
$ = {(2401)^{21}}$
$ = {(2402 - 1)^{21}}$
Now we use binomial theorem and expand the above function
${ = ^{21}}{C_0}{(2402)^0} \times {( - 1)^{21}}{ + ^{21}}{C_1}{(2402)^1} \times {( - 1)^{20}}{ + ^{21}}{C_2}{(2402)^2} \times {( - 1)^{19}} + .....{ + ^{21}}{C_{21}}{(2402)^{21}} \times {( - 1)^0}$
Find the value of the combination $^{21}{C_0}$
Therefore $^{21}{C_0} = \dfrac{{21!}}{{0!(21 - 0)!}}$
$ = \dfrac{{21!}}{{0! \times 21!}}$
$ = \dfrac{1}{{0!}}$
We know that $0! = 1$ , use this and we get
$ = 1$
Use this $^{21}{C_0} = 1$ and we get
$ = - 1{ + ^{21}}{C_1} \times 2402 + ....{ + ^{21}}{C_{21}}{(2402)^{21}}$
Except the first term, from second, all the terms are divisible by $2402$
We take all the divisible part as a constant $k$ , we get
$ = - 1 + 2402k$
Therefore the remainder when ${7^{84}}$ is divisible by $2402$ is $ - 1$ or $2402 - 1$ i.e., $2401$ .
The option (C) is correct.

Note: We know the property of power of a function ${a^{100}}$ , we can write this as ${({a^{10}})^{10}}$ . Using this property we can simplify any harder power of a function. A combination is a selection of items from a collection, such that the order of selection does not matter. The formula of the combination $^n{C_r} = \dfrac{{n!}}{{r!(n - r)!}}$ . We use binomial theorems to solve harder problems and expand a function.