Question

The last digit in \[{7^{300}}\] is:
A. 7
B. 9
C. 1
D. 3

Answer 1

Hint: We will first consider the given expression that is \[{7^{300}}\]. Then we will do the expansion of 7 in the first few terms and only consider the units digit and after the expansion of 4 to 5 terms we will see that the same pattern of units digit will be started so, as 300 is divisible by 4, we will check the unit digit at fourth term and that is the required answer.

Complete step by step answer:

We will first consider the expression given in the question that is \[{7^{300}}\].
Now, we will start expanding the terms of 7 till few terms,
Thus, we get,
\[
  {7^1} \to 7 \\
  {7^2} \to 9 \\
  {7^3} \to 3 \\
  {7^4} \to 1 \\
  {7^5} \to 7 \\
  {7^6} \to 9 \\
 \]
As we can see that terms have started repeating itself after 4 terms which implies that the trend repeats itself in multiples of 4.
Now, as we have to find the unit digit of \[{7^{300}}\] we will check whether the 300 is divisible by 4 or not because as the pattern gets repeated after 4 terms so, that’s why we will divide 300 by 4.
Thus, we get that 300 is divisible by 4 and the unit digit at 4th term is 1.
Thus, we can conclude that the unit digit of \[{7^{300}}\] is 1.
Hence, option C is correct.

Note: As we can not expand the terms till 300 terms so, we have determined that the expansion follows the pattern after a few terms. The unit digit of \[{7^2} = 49\] is 9 as it lies in the unit place and similarly, we have found the other unit digits. As the pattern repeats after 4 terms that is why we have to check whether 300 is divisible by 4 or not. open the powers of 7 properly. Have to check from where the pattern starts repeating.