For any integer ## a ##, the units digit of ## a^{2} ## is?

[Math100]

Homework Statement

Prove the following statement:
For any integer $a$, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$.

Relevant Equations

None.

Proof:

Let $a$ be any integer.
Then $a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8$, or $9\pmod {10}$.
Note that $a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4$, or $1\pmod {10}$.
Thus $a^{2}\equiv 0, 1, 4, 5, 6$, or $9\pmod {10}$.
Therefore, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$ for any integer $a$.

 

[Mark44]

Homework Statement:: Prove the following statement:
For any integer $a$, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$.
Relevant Equations:: None.

Proof:

Let $a$ be any integer.
Then $a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8$, or $9\pmod {10}$.
Note that $a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4$, or $1\pmod {10}$.
Thus $a^{2}\equiv 0, 1, 4, 5, 6$, or $9\pmod {10}$.
Therefore, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$ for any integer $a$.

Do you really need us to check this one for you? It's very simple.

 

[fresh_42]

Homework Statement:: Prove the following statement:
For any integer $a$, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$.
Relevant Equations:: None.

Proof:

Let $a$ be any integer.
Then $a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8$, or $9\pmod {10}$.
Note that $a^{2}\equiv 0, 1, 4, 9, 6, 5, 6, 9, 4$, or $1\pmod {10}$.
Thus $a^{2}\equiv 0, 1, 4, 5, 6$, or $9\pmod {10}$.
Therefore, the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$ for any integer $a$.

Sure, but what are units digits? I assume you meant unique digits. Units in algebra are usually elements that have a multiplicative inverse, e.g. $3\cdot 7 \equiv 1 \pmod {10}$ so $3^{-1}=7 \pmod {10}$ is a unit of
$$
\mathbb{Z}/10\cdot \mathbb{Z}=\mathbb{Z}_{10} = \{[0],[1],[2],[3],[4],[5],[6],[7],[8],[9]\}
$$
I have put the remainders in brackets because they only represent one possible system, even though a natural one. But we could as well take $\{[-4],[-3],[-2],[-1],[0],[1],[2],[3],[4]\}$ as system of remainders.

 

[Mark44]

Sure, but what are units digits? I assume you meant unique digits. Units in algebra are usually elements that have a multiplicative inverse

No, I'm sure he meant units digits - the decimal digits in the ones' place in the decimal representation of an integer. E.g., the units digit of 13 is 3. This has absolutely nothing to do with algebraic units or multiplicative inverses or any of that.

 

[Math100]

Sure, but what are units digits? I assume you meant unique digits. Units in algebra are usually elements that have a multiplicative inverse, e.g. $3\cdot 7 \equiv 1 \pmod {10}$ so $3^{-1}=7 \pmod {10}$ is a unit of
$$
\mathbb{Z}/10\cdot \mathbb{Z}=\mathbb{Z}_{10} = \{[0],[1],[2],[3],[4],[5],[6],[7],[8],[9]\}
$$
I have put the remainders in brackets because they only represent one possible system, even though a natural one. But we could as well take $\{[-4],[-3],[-2],[-1],[0],[1],[2],[3],[4]\}$ as system of remainders.

So $a\equiv 0, 1, 2, 3, 4, 5, 6, 7, 8$, or $9\pmod {10}\implies a^{2}\equiv 0, 1, 4, 9, 16, 25, 36, 49, 64$, or $81\pmod {10}$. This means the units digit of $a^{2}$ is $0, 1, 4, 5, 6$, or $9$ for any integer $a$.

 

[WWGD]

I'd say it comes down to showing that the units' digit of a alone determines the units' digit of $$a^2$$. So that you only need to check what happens with $$\{0,1,2,...9\}$$.
So maybe an induction argument of some sort would do.

 

[Mark44]

I'd say it comes down to showing that the units' digit of a alone determines the units' digit of $a^2$. So that you only need to check what happens with $\{0,1,2,...9\}$.

That's how I would do it.

So maybe an induction argument of some sort would do.

I don't see how this would work. There are only 10 digits, so it's straightforward to check them all to show what needs to be shown.

 

[WWGD]

That's how I would do it.

I don't see how this would work. There are only 10 digits, so it's straightforward to check them all to show what needs to be shown.

Sorry, I meant the number of digits. But I forgot the specifics of the argument I had. Edit: I think it had to see with the fact that in the decimal representation, all non-unit digits will end in 0, so that only the unit digits will make a difference in this respect.