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#1 2013-02-13 21:38:31
- niharika_kumar
- Novice
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exponents
units digit in the number (12357)^655
please tell me the solution!!
#2 2013-02-13 23:40:11
- bobbym
- Administrator
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Re: exponents
Hi;
Welcome to the forum. It is a 3.
Answer is 3.
Last edited by bobbym (2013-02-13 23:47:27)
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#3 2013-02-14 10:44:03
- noelevans
- Full Member
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Re: exponents
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.
#4 2013-02-14 12:12:09
- cooljackiec
- Full Member
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Re: exponents
is noelevans levans from AoPS?
I see you have graph paper.
You must be plotting something
#5 2013-02-14 17:21:55
- bobbym
- Administrator
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Re: exponents
Hi;
I do not think so. Noel is polite.
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#6 2013-02-16 18:53:02
- niharika_kumar
- Novice
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Re: exponents
thnx guys,
epecially noel as m a student of class 9 and thus his soln ws much more easy fr me:).
#7 2013-02-18 09:49:10
- noelevans
- Full Member
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Re: exponents
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.
#8 2013-02-18 10:10:18
- anonimnystefy
- Real Member
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Re: exponents
Hi noelevans
AoPS is another forum. It's full name is Art of Problem Solving.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#9 2013-02-18 10:52:30
- noelevans
- Full Member
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Re: exponents
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.
#10 2013-02-18 11:02:37
- anonimnystefy
- Real Member
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Re: exponents
Hi
You're welcome (I think)!
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#12 2013-02-19 11:24:48
- anonimnystefy
- Real Member
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Re: exponents
Depends on what you consider good.
The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“A secret's worth depends on the people from whom it must be kept.” ― Carlos Ruiz Zafón
#13 2013-02-19 12:16:04
- bobbym
- Administrator
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Re: exponents
You think therefore you am.
In mathematics, you don't understand things. You just get used to them.
90% of mathematicians do not understand 90% of currently published mathematics.
I am willing to wager that over 75% of the new words that appeared were nothing more than spelling errors that caught on.
#14 2013-02-19 12:24:19
- mathgogocart
- Super Member
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Re: exponents
bobbym wrote:
You think therefore you am.
such wise of a post.
woosh! woosh! bye as I go to Kanto.