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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,062

units digit in the number (12357)^655

please tell me the solution!!

friendship is tan 90°.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,737

Hi;

Welcome to the forum. It is a 3.

Answer is 3.

*Last edited by bobbym (2013-02-13 00:47:27)*

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Or another way to look at it:

0th power of 12357 ends in 1.

1st power ends in 7. 2nd power ends in 9. 3rd power ends in 3. 4th power ends in 1

5th power ends in 7 6th power ends in 9 7th power ends in 3. 8th power ends in 1

9 ... 7 10 ... 9 11 ... 3 12 ... 1, etc.

There are only four digits that occur in the powers and these repeat in the sequence 1, 7, 9, 3, 1, ...

Divide the power (655 in this case) by four and see what the remainder is. It is three. So the 655th

power ends in the same digit that the 3rd power does, namely 3.

But this is a bit confusing since we get 3 for the remainder and for the last digit.

Try raising to the 656 power. Then dividing 656 by 4 we get a remainder of zero. So the 656

power ends in 1 same as the 0th power.

For the 654th power dividing 654 by 4 has remainder 2. So the 654th power ends in 9 the same

as the second power.

For the 653rd power dividing 653 by 4 has remainder 1 so the units digit of the 653rd power is 7 the

same as the 1st, 5th, 9th, ... powers.

Be very blesses!

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**cooljackiec****Member**- Registered: 2012-12-13
- Posts: 185

is noelevans levans from AoPS?

I see you have graph paper.

You must be plotting something

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,737

Hi;

I do not think so. Noel is polite.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**niharika_kumar****Member**- From: Numeraland
- Registered: 2013-02-12
- Posts: 1,062

thnx guys,

epecially noel as m a student of class 9 and thus his soln ws much more easy fr me:).

friendship is tan 90°.

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Hi niharika_kumar,

You are quite welcome! Seeing patterns is often the name of the game in solving math problems.

And if you have access to a good math program you can often write a program to test for patterns

associated with a given problem.

Hi cooljackiec, Not only am I not levans, I have no clue as to what AoPS is. What is it?

And thanks bobbym for the kind comment. I do try to be polite and not hurt anyone's feelings

even if I think what they are saying is incorrect. After all, I have been wrong before. As I

recall once in 1947 I was wrong about ...

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Hi noelevans

AoPS is another forum. It's full name is Art of Problem Solving.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Thanks stefy!

I might have worried all night about what that meant. I'll check the site out and see what it is about.

Bless you!

Writing "pretty" math (two dimensional) is easier to read and grasp than LaTex (one dimensional).

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Hi

You're welcome (I think)!

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**noelevans****Member**- Registered: 2012-07-20
- Posts: 236

Yes, you do think. And you are quite good at it!

LaTex is like painting on many strips of paper and then stacking them to see what picture they make.

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**anonimnystefy****Real Member**- From: Harlan's World
- Registered: 2011-05-23
- Posts: 16,016

Depends on what you consider good.

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

The knowledge of some things as a function of age is a delta function.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 105,737

You think therefore you am.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.****No great discovery was ever made without a bold guess. **

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**Mathegocart****Member**- Registered: 2012-04-29
- Posts: 1,667

bobbym wrote:

You think therefore you am.

such wise of a post.

The integral of hope is reality.

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