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  Discussion about math, puzzles, games and fun.   Useful symbols: √ ∞ ≠ ≤ ≥ ≈ ⇒ ∈ Δ θ ∴ ∑ ∫ π -




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Topic review (newest first)

2013-02-19 12:24:19

bobbym wrote:

You think therefore you am.

such wise of a post.

2013-02-19 12:16:04

You think therefore you am.

2013-02-19 11:24:48

Depends on what you consider good.

2013-02-19 11:08:00

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

2013-02-18 11:02:37


You're welcome (I think)!

2013-02-18 10:52:30

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!  smile

2013-02-18 10:10:18

Hi noelevans

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

2013-02-18 09:49:10

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. smile

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 ... smile

2013-02-16 18:53:02

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

2013-02-14 17:21:55


I do not think so. Noel is polite.

2013-02-14 12:12:09

is noelevans levans from AoPS?

2013-02-14 10:44:03

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! smile

2013-02-13 23:40:11


Welcome to the forum. It is a 3.

Answer is 3.

2013-02-13 21:38:31

units digit in the number (12357)^655

please tell me the solution!!dunno

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