1. ## need help

prove that any number that is a square must have one of the following for its units digit: 0,1,4,5,6,9.

2. Originally Posted by mancillaj3
prove that any number that is a square must have one of the following for its units digit: 0,1,4,5,6,9.
The squares modulo 10 are: 0,1,4,5,6,9.

3. Consider a two digit number with unit digit as a and tens digit as b
So a and b both are positive interger less than 10
So number can be written as 10b + a

Now $(10b + a)^2 = 100b^2 + 20ab + a^2$

Now $100b^2$ and $20ab$ both have 0 at unit place. So unit digit of $100b^2 + 20ab + a^2$ will be decided by $a^2$

And for a being single digit integer, you only get 0,1,4,5,6,9.at units place of $a^2.$

You can extend this proof for three digit number 100c + 10b + a.
Same logic will be applied in this case