# An easier way to find out the unit digit of a number indexed with a positive integer using Modular A

In many of competitive exams, you are asked to find out unit digit in some n^p where n and p are positive integers. Sometimes it is a lengthy process to find out the correct answer, and remember one principle, avoid unnecessary multiplications. Hence given below is the method using Modular Arithmetic where less and less calculations have been used.

Let us begin with an example. What is the unit digit in 2^103?

If you do it with traditional method, you know that the unit digit in 2 raised to some index repeats itself every 4th time. Means; 2^1 = 2; 2^5 = 32 (unit digit is 2); 2^9 = 512 (unit digit is 2); 2^13 = 8192 (unit digit is 2). Now nth term of this AP (1, 5, 9, 13, …) is 1+(n-1)4 = 4n-3, and for n = 26, we have 101.

Thus, the unit digit in 2^101 = 2; and unit digit in 2^2 = 4.

Hence, unit digit in 2^103 = 2x4 = 8.

Now look at the solution from a different perspective:

For any number indexed with any positive integer, i.e., n^p, the unit number shall be the remainder when this number n^p is divided by 10. And we shall do it using Modular Arithmetic.

For example, unit digit in 3^7 is equal to remainder when 3^7 = 2187 is divided by 10, i.e. 7.

Let us do some questions. It is easy, less time consuming and less cumbersome: The questions with solutions are here .......