Question

The remainder we get when we divide ${27^{40}}$by 12 is
A)3
B)7
C)9
D)11

Answer 1

Hint:Solve this problem by using the concept of Modulo arithmetic and congruence rules which is if a-b is divisible by n we write it as $a \equiv b(\bmod n)$.The formulae that will be useful in solving this sum will be
If $a \equiv b(\bmod n)$ then ${a^m} \equiv {b^m}(\bmod n)$;
If $a \equiv b(\bmod n)$ and $b \equiv c(\bmod n)$then $a \equiv c(\bmod n)$.

Complete step by step answer:
We know
$27 \equiv 3(\bmod 12)$;
Now by using property that is written in hint ,which is
If $a \equiv b(\bmod n)$ then ${a^m} \equiv {b^m}(\bmod n)$; …………….Eq(1)
We get ${27^{40}} \equiv {3^{40}}(\bmod 12)$,which is equal to ${27^{40}} \equiv {9^{20}}(\bmod 12)$……………..Eq(2)
Now we will try to convert ${3^{40}}$ into further lower number by which we can find the remainder easily.
We can write ${3^{40}}$ =\[{({3^2})^{20}}\]=${9^{20}}$
We know that
$9 \equiv - 3(\bmod 12)$ …….………Eq(3)
By using Eq(1) and Eq(3) we get
${9^{20}} \equiv {( - 3)^{20}}(\bmod 12)$ …………….Eq(4)
We can write \[{( - 3)^{20}} = {({( - 3)^2})^{10}} = {9^{10}}\] …………….Eq(5)
By using Eq(4) and Eq(5) we get
${9^{20}} \equiv {9^{10}}(\bmod 12)$ ..………….Eq(6)
We also know that If $a \equiv b(\bmod n)$ and $b \equiv c(\bmod n)$then $a \equiv c(\bmod n)$………….Eq(7)
 Which implies \[{27^{40}} \equiv {9^{10}}(\bmod 12)\](by using Eq (2),(5),(7)) …………..Eq(a)
By using Eq(1) and Eq(3), we get
${9^{10}} \equiv - {3^{10}}(\bmod 12)$ …………….Eq(8)
We can write \[{( - 3)^{10}} = {({( - 3)^2})^5} = {9^5}\] …………….Eq(9)
Which implies \[{27^{40}} \equiv {9^5}(\bmod 12)\](by using Eq (a),(9),(7)) …………..Eq(b)

By using Eq(1) and Eq(3), we get
\[{9^5} \equiv - {3^5}(\bmod 12)\] …………….Eq(10)
We know that \[ - {3^5}\]=-243 and \[ - 243 \equiv 9(\bmod 12)\] …………….Eq(11)
By using Eq(b),Eq(10),Eq(11) in Eq(7)
We get,

\[{27^{40}} \equiv 9(\bmod 12)\]
Which means if 9 is subtracted from ${27^{40}}$ then it will be divisible by 12.
So by using the basic principle of division,
This implies when ${27^{40}}$is divided by 12 we get the remainder as 9.
So the correct option is C.

Note:
Do not make any assumptions while solving this problem. Learn all the properties of congruence which is very important while solving these types of problems. Do not make mistakes while taking negative and positive integers during the congruency that is in mod form.