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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.

Jul 21, 2020

#1
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The ten's digit can be any odd number.

Jul 21, 2020
#3
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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
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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  =  100a2 + 20ab + b2

The  100a2  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 b2 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
#5
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Wow! This makes sense, thank you so much! I really understand it now! Guest Jul 22, 2020