Can you find out the remainder when 3^300 is divided by 5?

3

It is obvious that it is not feasible to calculate 3^300 as it will take too much of time. So we will use a trick to solve the question. We will calculate the remainder of each power till we find a pattern.

3^1 divided by 5 leaves the remainder 3.

3^2 divided by 5 leaves the remainder 4.

3^3 divided by 5 leaves the remainder 2.

3^4 divided by 5 leaves the remainder 1.

3^5 divided by 5 leaves the remainder 3.

3^6 divided by 5 leaves the remainder 4.

As you can see that the pattern is now repeating itself and it will go on like this till 3^300 and beyond. Since every fourth remainder is same as the first, we will look for the power of 4 only. 300 is divisible by 4. Therefore at the power of 300, the first remainder will repeat itself and the remainder will be 3.