Question

The digit in the unit place of ${{7}^{291}}$ is:
(a) 1
(b) 2
(c) 3
(d) 4

Answer 1

Hint: Find the pattern of the repetition of the unit digit of ${{7}^{n}}$ by putting some integer values of $n$. Once the value of $n$ is obtained, divide 291 by this value of $n$. The remainder obtained will be a small value and the unit digit can be further determined by simple multiplication.

Complete step-by-step answer:

The unit digit of any number raised to the power something, repeats itself after a certain interval of power. This forms a pattern of repetition of unit digit which is used to solve the question.

Now, let us come to the question.
${{7}^{1}}$$=7$, has 7 as its unit place digit.
${{7}^{2}}=49$, has 7 as its unit place digit.
${{7}^{3}}=343$, has 3 as its unit place digit.
${{7}^{4}}=2401$, has 1 as its unit place digit.

Now, the unit place digit has become 1, so the unit place will start to repeat itself. You can see that ${{7}^{5}}$ will have 7 as its unit place digit.
So, the value of $n$ is 4. That means the unit digit of ${{7}^{n}}$ will repeat itself after every multiple of 4.

Now, ${{7}^{291}}$ can be written as, ${{7}^{291}}={{\left( {{7}^{4}} \right)}^{72}}\times {{7}^{3}}$. Here, ${{7}^{4}}$ will have 1 as its unit digit. Any number having 1 as its unit digit is when raised to any power, the unit digit remains 1. So, ${{\left( {{7}^{4}} \right)}^{72}}$ will have its unit digit equal to 1. Now, ${{7}^{3}}$ has 3 at its unit place. Therefore, ${{\left( {{7}^{4}} \right)}^{72}}\times {{7}^{3}}$ will have its unit digit, 3 multiplied by 1, that is 3.

Hence, option (c) is the correct answer.

Note: As you can see that ${{7}^{291}}$ is a huge number, so, it is not possible for us to multiply 7 that number of times to get the unit place digit. It is therefore necessary to find a better and easy approach so that the question can be solved in less time. To do this, we just have to multiply 7 the number of times till we observe the pattern of repetition.