How to find a Square-root, by hand.

There are a number of methods to find a square root, factoring it might work but there are no trivial factors of 5186736361. However an alternate to find a square root, without a calculator is to do the following.

Lets work with an example 5186736361, as this is bigger than a 2³² it's not immediately easy to find the square-root manually (without the aid of a calculator).

Proceed as follows, break the number into pairs of digits starting from the right 51 68 73 63 61

Now start with the first pair

what 'x' * 'x' is <= 51 (where 'x+1' * 'x+1' is > 51). Easy, 7

Now double the '7' at the left, bring down the next two digits (86) and repeat. What is 'x' that 14x * x <= 286 (where x is a digit such that if x was 0 the number would be 140*0 = 0, if x was 1 the number would be 141*1=141 etc.)

Clearly x = 2 in this case

Bring down the 142 on the left, doubling the 2 to make it 144. Bring down the next two digits on the right (73) and repeat the questions. What is 'x ' such that 144x * x <= 237?

That's easy, it's got to be 0.

Being down the 63, double the 0 and add it to 144. What is 'x' such that 1440x * x <= 27636?

x would be one.

Double the 1 and add it to the end of the 1440 (to give 14402). Bring down the 61, now what is 'x' such that 14402x * x <= 1296261?

This one is not quite so simple, however a single multiple shows that 9 * 144029 = 1296261.

Thus the square root of 5186736361 is 72019.