Question

The digit in the unit’s place of ${\left( {141414} \right)^{12121}}$ is
A) 4
B) 6
C) 3
D) 1

Answer 1

Hint:
First of all, we will divide the index with 4 and then we will re – write the given number in terms of the quotient and the remainder of the index power. Then, we will check the digit at unit’s place for various powers of 4 and hence, we will calculate the value of the digit at unit’s place in ${\left( {141414} \right)^{12121}}$.

Complete step by step solution:
We are required to find the digit in unit’s place of ${\left( {141414} \right)^{12121}}$
Since the unit digit of 141414 is 4, we will divide the index with 4 for the remainder.
Upon dividing 12121 with 4, we get quotient as 303 and the remainder is 1.
Now, we can write the index using the division algorithm as: dividend = quotient$ \times $ divisor + remainder (i.e., a = bq + r) as: $12121 = 303\left( 4 \right) + 1$
Therefore, ${\left( {141414} \right)^{12121}}$= ${\left( {141414} \right)^{303\left( 4 \right) + 1}}$
It can also be written as: ${\left( {141414} \right)^{303\left( 4 \right) + 1}} = {\left( {{{141414}^4}} \right)^{303}}{\left( {141414} \right)^1}$
Now, we only need the unit place of 141414, therefore, we can write the above equation as: ${\left( {{4^4}} \right)^{303}}{\left( 4 \right)^1}$
Now, the unit digits in various powers of 4 can be written as: ${4^1} = 4$ , ${4^2} = 6$ , ${4^3} = 4$ , ${4^4} = 6$ and so on. Hence, we can write the equation ${\left( {{4^4}} \right)^{303}}{\left( 4 \right)^1}$ as ${\left( 6 \right)^{303}}\left( 4 \right)$.
Now, we know that for every power of 6, the unit digit is always 6. Hence, we can write ${\left( 6 \right)^{303}}\left( 4 \right)$ as $6 \times 4 = 24$ and the digit at unit place is 4.

Hence option (A) is correct.

Note:
In such questions, being confused is quite easy at places after we have used a division algorithm since we need to calculate the digit at unit place, therefore we have used the unit digit of 141414 to proceed further. You may go wrong while considering the digit at one place in the powers of 4 since we were required to place the digit at unit place in ${4}^{\text{th}}$ power of 4.