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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

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

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Hi;

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

Could you please explain? Like how you did it.

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

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

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

Mod

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

Yes

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

It always end with 5. But, why?

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

5. But why it always end with 5?

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

I see, I understand that know.

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

1+5 = 6 And I know that :

6^1 = 6

6^2 = 36

6^6 = 216

.... Then it's 6

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

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

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

Where can we not put a six?

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

The answer

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

So the answer is 6? Written format please.

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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**bobbym****bumpkin**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 109,606

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.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.**** Always satisfy the Prime Directive of getting the right answer above all else.**

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**David****Member**- From: Bumpkinland
- Registered: 2014-04-23
- Posts: 3,137

Can we learn that? Please? (Congruences)

Meaningless, meaningless, Everything is meaningless! - Ecclesiastes 1:2

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