Square root of perfect squares ending in 25

What this question given in the primary category of the 1^st International Vedic Mathematics Olympiad in September 11, 2021 really requires is to find out the square root of of 5625. Anyone who is familiar with Vedic Mathematics can answer this “By Mere Observation” which was the sutra we discussed in our two previous posts.

But even those who have no knowledge of Vedic Math or who do not know how to extract the square root of perfect squares may answer this question without resorting to the “guess and check” or “trial and error” method.

First, since the number 5625 ends in 5, the last digit of the square root must be 5. This is confirmed because all the choices involve numbers ending in 5.

Next if we look at the first two digits of the perfect square, 56, we can see the it is more than 49 which is the square of 7 but less that 64 which is the square of 8.

This means that the square root of 5625 lies between 70 and 80 and with 5 as the last digit. Therefore, D) 75 x 75 is the answer.

In fact, 56 is 7 x 8, which follows the Vedic Math formula for squaring numbers ending in 5: the first part is obtained “By One More than the One Before” and the second part is always 5 x 5 or 25. In 75 x 75, the number before 5 is 7 and one more than 7 is 8. Thus 75 x 75 = 7 x 8 | 5 x 5 = 56| 25 = 5625.

Exercises: Find the square root of the following:

1. 625
2. 7225
3. 9025
4. 225
5. 3025
6. 2025
7. 11025
8. 15625
9. 13225
10. 990,025

Answers to previous exercises: The remainder when the number is divided by 11.

1. 1334 → 3
2. 3777 → 4
3.  2323 → 2
4. 5678 → 2
5. 12345 → 3
6. 54321 → 3
7. 25552 → 10
8. 235,238 → 3
9. 238,235 → 8
10. 555,222 → 8