Find the unit digit of the number n

David · 2014-06-12 20:48:40

Hint 5^4 = 5 x 5 x 5 x 5.

bobbym · 2014-06-12 20:57:53

Hi;

David · 2014-06-12 20:59:08

Could you please explain? Like how you did it.

bobbym · 2014-06-12 21:02:41

Mods can be split up for computational purposes. Look at the first one:

(1 + 5^2000)

you know that 5^2000 must end in a 5 so the last digit of that term is 6. As a matter of fact the last digit of each of those terms is a 6. We have 8 of them which is 6^8 but we know that each multiple of 6 ends in a 6 so the whole expression ends in a 6.

David · 2014-06-12 21:10:06

Thanks, bobbym but I still don't get it so e reading resource would be great!

bobbym · 2014-06-12 21:13:46

Which part did you not understand? I will break it down further.

David · 2014-06-12 21:14:58

Mod

bobbym · 2014-06-12 21:18:51

The question depends on mods. You see when they say the last digit they are really saying what is the number mod 10. You will have to learn them.

But I will show you how to do the problem in a slightly less rigorous way that will get the right answer. Ready?

David · 2014-06-12 21:36:04

Yes

bobbym · 2014-06-12 21:46:05

We start with the first term

(1 + 5^2000)

we want the last digit of that. Do a few experiments

5^1 = 5
5^2 = 25
5^3 = 125
5*4 = 625
5^5 = 3125

hmmm , what is the last digit of 5^2000?

David · 2014-06-12 21:49:50

It always end with 5. But, why?

bobbym · 2014-06-12 21:50:36

Did you see that they all end in 5? Without understanding the reason why that is true if I asked you to guess what the last digit of 5^2000 is what would you try.

David · 2014-06-12 21:51:39

5. But why it always end with 5?

bobbym · 2014-06-12 21:54:59

Yes, 5 makes sense.

But why it always end with 5?

I can answer that by asking you to look at your hint in post #1.

5^3 = 5*5^2

since 5^2 ended in a 5 and 5 x 5 = 25 therefore 5^3 ends in a 5 too.

David · 2014-06-12 21:56:26

I see, I understand that know.

bobbym · 2014-06-12 21:57:50

So if we know that 5^2000 ends in a 5 what does 1 + 5^2000 end in?

David · 2014-06-12 21:59:12

1+5 = 6 And I know that :
6^1 = 6
6^2 = 36
6^6 = 216
.... Then it's 6

bobbym · 2014-06-12 22:01:58

Your problem gets to look like this

(1+5)(1+5)(1+5)(1+5)(1+5)(1+5)(1+5)(1+5) = 6 x 6 x 6 x 6 x 6 x 6 x 6 x 6 which ends in a 6. We are done.

David · 2014-06-12 22:03:40

But we can't put six there right? And btw could you show me the whole written format? Ty

bobbym · 2014-06-12 22:04:49

Where can we not put a six?

David · 2014-06-12 22:05:23

The answer

bobbym · 2014-06-12 22:08:16

Each parentheses reduces down to a single 6. There are 8 of them that are multiplied together. But you know that every 6 when multiplied by another 6 produces a 6. Therefore that product of 6's has a 6 on the end.

David · 2014-06-12 22:09:21

So the answer is 6? Written format please.

bobbym · 2014-06-12 22:13:35

You could write down exactly what I have written in the posts above. It is correct. If you want something else we have to go to congruences.

David · 2014-06-12 22:14:17

Can we learn that? Please? (Congruences)

Math Is Fun Forum

#1 2014-06-12 20:48:40

Find the unit digit of the number n

#2 2014-06-12 20:57:53

Re: Find the unit digit of the number n

#3 2014-06-12 20:59:08

Re: Find the unit digit of the number n

#4 2014-06-12 21:02:41

Re: Find the unit digit of the number n

#5 2014-06-12 21:10:06

Re: Find the unit digit of the number n

#6 2014-06-12 21:13:46

Re: Find the unit digit of the number n

#7 2014-06-12 21:14:58

Re: Find the unit digit of the number n

#8 2014-06-12 21:18:51

Re: Find the unit digit of the number n

#9 2014-06-12 21:36:04

Re: Find the unit digit of the number n

#10 2014-06-12 21:46:05

Re: Find the unit digit of the number n

#11 2014-06-12 21:49:50

Re: Find the unit digit of the number n

#12 2014-06-12 21:50:36

Re: Find the unit digit of the number n

#13 2014-06-12 21:51:39

Re: Find the unit digit of the number n

#14 2014-06-12 21:54:59

Re: Find the unit digit of the number n

#15 2014-06-12 21:56:26

Re: Find the unit digit of the number n

#16 2014-06-12 21:57:50

Re: Find the unit digit of the number n

#17 2014-06-12 21:59:12

Re: Find the unit digit of the number n

#18 2014-06-12 22:01:58

Re: Find the unit digit of the number n

#19 2014-06-12 22:03:40

Re: Find the unit digit of the number n

#20 2014-06-12 22:04:49

Re: Find the unit digit of the number n

#21 2014-06-12 22:05:23

Re: Find the unit digit of the number n

#22 2014-06-12 22:08:16

Re: Find the unit digit of the number n

#23 2014-06-12 22:09:21

Re: Find the unit digit of the number n

#24 2014-06-12 22:13:35

Re: Find the unit digit of the number n

#25 2014-06-12 22:14:17

Re: Find the unit digit of the number n

Board footer