The units digit of a perfect square is 6. What are the possible values of the tens digit?

I am aware this was posted before by Mellie. However, the question got no answers!

Please help me!!! Thank you so much!! A solution that is fully explained would be really helpful!

Also, I figured out that the units digit for the number we are squaring is 4 or 6, but I can't really progress on the tens digit.

Guest Jul 21, 2020

#1

#3**+1 **

Okay, by how did you get that? I was squaring some of the numbers, and I noticed that they were odd, but do you know how to prove that?

Guest Jul 22, 2020

#4**+1 **

The unit's digit must be either a 4 or a 6 for the square to end in 6; all other possibilites can be excluded by trial.

If you are squaring a two-digit number, you can represent this number as 10a + b (knowing that b must be

either 4 or 6).

(10a + b)^{2} = 100a^{2} + 20ab + b^{2}

The 100a^{2} can be ignored for it does not affect the ten's digit.

20ab must be even; an even number, 20, times either an even number or an odd number results in an even number.

To this term (20ab) must be added the carry of the b^{2} term.

If b = 4, the carry is 1.

If b = 6, the carry is 3.

Adding an odd number (either 1 or 3) to an even number (20ab) results in an odd number.

By trying various possibiliteis, you will discover that the ten's digit can be any of the odd digits.

geno3141
Jul 22, 2020