Detemine the unit digit of a number

Question

Find the unit digit of the number: $$3^{7005} \times 6^{8000}$$ My turn: $$3^{7005}\times 6^{8000}=3^{7005}\times 3^{8000} \times2^{8000}$$ $$3^{13005} \times 2^{8000}$$ But i could not go on ?

Well, you can start by the fact that the last digit of $6^n$ is always $6$ for any positive whole number $n$. Then you only have the $3^{7005}$ to deal with. — Matti P., Mar 08 '19 at 13:39

José Carlos Santos · Answer 1 · 2019-03-08T14:14:45.520

4

The unit digit of the numbers of the powers of $3$ form a cyclic sequence: $1,3,9,7,1,3,9,7,\ldots$ In particular, the unit digit of $3^n$ is $3$ if $n$ is of the form $4k+1$. And the unit digit of any power of $6$ is $6$. Since $7\,005$ is of the form $4k+1$, the answer to your question is $8(=3\times6\pmod{10})$.

edited Mar 08 '19 at 14:14

answered Mar 08 '19 at 13:39

José Carlos Santos

427,504

Agreed! I wonder why someone downvoted ... – Matti P. Mar 08 '19 at 13:50
It happens all the time: :-( – José Carlos Santos Mar 08 '19 at 13:51
Maybe the down-voter, like me, can't parse the sentence beginning with "In particular..."? Oh, I just got it. "of" is "if". – B. Goddard Mar 08 '19 at 14:08
@B.Goddard Oops! I've edited my answer. Thank you. – José Carlos Santos Mar 08 '19 at 14:15

Brian61354270 · Answer 2 · 2019-03-08T14:53:08.320

2

We want the unit digit of the number $3^{7005} \cdot6^{8000}$. This is equivalent to asking for the residue of the number $3^{7005} \cdot6^{8000}$ modulo $10$. Hence, we want some number $n$ with $0 \leq n \leq 9$ such that $$ n \equiv 3^{7005} \cdot6^{8000} \pmod{10} $$

Note that $3^4 \equiv 1 \pmod{10}$ and $6^n \equiv 6 \pmod{10}$ for all $n\in\mathbb{N}$. With this result, we have \begin{align} 3^{7005} \cdot6^{8000} &= 3^{7004}\cdot 3^{1} \cdot 6^{8000} \\ &= \left(3^{4}\right)^{1751}\cdot 3^{1} \cdot 6^{8000}\\ &\equiv1^{1751}\cdot 3^{1} \cdot 6^{1} \pmod{10} \\ &= 3 \cdot 6 \\ &\equiv 8 \pmod{10} \\ \end{align} Hence, the unit digit of $3^{7005} \cdot6^{8000}$ is $8$.

edited Mar 08 '19 at 14:53

answered Mar 08 '19 at 14:12

Brian61354270

2,170

1

Right outcome, slight error in reasoning. $3^4\equiv 1 \mod 10;\ 3^7\equiv 7 \mod 10$. You figured it out right when you found $3^5\equiv 3 \mod 10$. Note that $4\mid 7004$. – Keith Backman Mar 08 '19 at 14:50
@KeithBackman I'm shocked I overlooked that. I've edited my response with your correction. Thanks for reviewing my answer so closely! – Brian61354270 Mar 08 '19 at 14:54
Not to worry -- you should see some of the bloopers I've made! – Keith Backman Mar 08 '19 at 15:56

score 1 · Answer 3 · edited Mar 08 '19 at 13:52

1

If you think about the way you typically do multiplication, the unit digit of a product is the product of the unit digits.

Can you figure out the unit digit of the two terms in the product?

Hint: calculate $3^0, 3^1, 3^2, 3^3, 3^4, 3^5,\ldots$ Do you see a pattern in the unit digits? Do you see why the pattern holds?

edited Mar 08 '19 at 13:52

NickD

2,097

answered Mar 08 '19 at 13:41

Daniel McLaury

24,656

Detemine the unit digit of a number

3 Answers3