Find the last digit of 7^2007

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Mathlover
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Find the last digit of 7^2007

Unread post by Mathlover » Fri Jul 01, 2011 4:51 pm

my answer is 8,is it correct??

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nayel
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Re: Find the last digit of 7^2007

Unread post by nayel » Fri Jul 01, 2011 11:47 pm

Your answer is incorrect, because $7^{2007}$ is clearly an odd number and so its last digit must be odd. How did you find it?
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

Mathlover
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Re: Find the last digit of 7^2007

Unread post by Mathlover » Sat Jul 02, 2011 3:55 pm

can u plz show me the solution??

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Masum
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Re: Find the last digit of 7^2007

Unread post by Masum » Sat Jul 02, 2011 5:03 pm

Mathlover wrote:my answer is 8,is it correct??
$2007\equiv3\pmod4$ and the last digit of $7^3$ is $3$, so the answer is $3$.
Think why it works.
One one thing is neutral in the universe, that is $0$.

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nayel
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Re: Find the last digit of 7^2007

Unread post by nayel » Sat Jul 02, 2011 8:33 pm

@Masum: Why don't you stop ruining the problems for them and let them solve the problems?
Mathlover wrote:can u plz show me the solution??
Can you find the remainder when $7^{2007}$ is divided by $10$?
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

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Masum
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Re: Find the last digit of 7^2007

Unread post by Masum » Mon Jul 04, 2011 1:19 pm

nayel wrote:@Masum: Why don't you stop ruining the problems for them and let them solve the problems?
For their thinking why it should work. This would lead them to the solution as well as a new idea.
One one thing is neutral in the universe, that is $0$.

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nayel
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Re: Find the last digit of 7^2007

Unread post by nayel » Mon Jul 04, 2011 8:17 pm

How do you even know he/she knows what mod is?
"Everything should be made as simple as possible, but not simpler." - Albert Einstein

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Moon
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Re: Find the last digit of 7^2007

Unread post by Moon » Mon Jul 04, 2011 10:51 pm

Okay. I personally think that the best idea is to give them some hints (for the easy problems)/hide the solution. However, when some relatively problem is posted you may choose to post solution normally.

BTW hiding is really easy. Just try

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 [h] your text inside [/h] 
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Tahmid Hasan
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Re: Find the last digit of 7^2007

Unread post by Tahmid Hasan » Tue Jul 05, 2011 11:52 am

$7^2 \equiv -1(mod 10) $
or,$7^{2006} \equiv -1(mod 10) $
it follows $7^{2007} \equiv -7(mod 10)$.so,$7^{2007} \equiv 3(mod 10)$
so the last digit is $3$
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Akash
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Re: Find the last digit of 7^2007

Unread post by Akash » Sat Jul 16, 2011 11:42 pm

the problem can be solved without using modular arithmetic. the last digit of 7^1 is 7, for 2 its 9, for 3 its 3, for 4 its 1 and for 5 its again 7.as we can write 2007 like that ….4*501+3……the last digit of 7^2007 will be the last digit of 7^3, no need of mod.

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