Best Answer Ever
Compute
3 2 0 1 4 ( m o d 1 0 0 )
without the use of a calculator or other aids.
The answer is 69.
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9 solutions
Perfect. Who else got the joke in the title? :D
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Oh, risque, Hulse.
Moi, ze ansa waz verry punny. :D
Mandatory Xkcd refference: xkcd 487
I don't understand. Where is the joke?
Hey, @Ankit Vijay had 69 followers as of June 3, 2014 10:30 PT.
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You're right I'm 13. :D
U mean d perverted meaning of 69?!
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Ahem ahem. XD
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@Yuxuan Seah – Lol 69 XD
me!!!!!!!!!!!!!!!!!!!!!!!!!!!
don't you think that this place is meant for PCM? be in control, please!
Did anybody actually calculate phi(100) by hand... or is it okay to just google it?
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ϕ ( p 1 α 1 p 2 α 2 p 3 α 3 ⋯ p n α n )
= ( p 1 α 1 − p 1 α 1 − 1 ) ( p 2 α 2 − p 2 α 2 − 1 ) ⋯ ( p n α n − p n α n − 1 )
,where p 1 , p 2 , … , p n ∈ P , α 1 , α 2 , … , α n ∈ N + .
We thus have ϕ ( 1 0 k ) = 4 ⋅ 1 0 k − 1 , ∀ k ∈ N + .
You can calculate it as 1 0 0 ( 1 − 1 / 2 ) ( 1 − 1 / 5 ) = 4 0 .
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It won't be twenty, based off of Euler's Totient. Look at the solution above. And how do you know that it is 3 2 0 ?
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Last two digits of 3 1 0 = 4 9 . So last two digits of 3 2 0 will be 4 9 2 = 01.
So it must be 20.
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@Vinay Sipani – You have only shown that 3 2 0 ≡ 3 4 0 ≡ 3 6 0 ≡ … ≡ 3 2 0 0 0 ≡ 1 ( m o d 1 0 0 )
Something tells me that you don't get the joke AND you can't use Euler's totient function properly. @Vinay Sipani
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yes...its true...I have never read about Euler's totient .I can just imagine what it is...
Can you make it clear please what is phi, here please?
hahahah !!
Why is it mod 10?Shouldn't it be mod 100?Someone pls help....
3 2 0 1 4 = 9 1 0 0 7 = ( 1 0 − 1 ) 1 0 0 7 Expand it using binomial to get the answer.
great method
Usually it's lengthy with bino but this one is the simplest.... Good job.
3 2 0 1 4 = ( 3 2 ) 1 0 0 7 = ( 1 0 − 1 ) 1 0 0 7 = 1 0 0 × A + 1 0 × C 1 0 0 7 1 0 0 6 − C 1 0 0 7 1 0 0 7 3 2 0 1 4 = 1 0 0 × A + 1 0 0 6 9 ⇒ 3 2 0 1 4 ( m o d 1 0 0 ) = 6 9
Thanks for explaining my solution!!
I could not understand the joke in the title and I did it using the long method
We have to find what's the remainder when 3^2014 is divided by 100.
3^10 = 49 mod 100 (use calculator while violating the rules;) or do it by just multiplying the last 2 digits of the last term(3^1 = 3, 3^2 = 9,.... 3^10 = 49 as last 2 digits))
Multiplying the powers of both sides by 10,
3^10*10 = 49^10 mod 100
3^100 = 49^10 mod 100 (49^10 gives last 2 digits as 01, this can be found easily)
3^100 = 1^10 mod 100
Multiplying the powers of both sides by 20,
3^2000 = 1^200 mod 100 == 3^2000 = 1 mod 100------------------------------1
Also, 3^14 = 69 mod 100-------------------------------2
Multiplying 1 and 2,
3^2014 = 69 mod 100
I think that I lengthened it a little, sorry for that but just the simplest solution, I think.
You just told @Krishna Ar that 3 2 0 1 2 is 41 mod 1000
So, I just multiplied 9
:)
LOL! But who told me?
@Finn told you I think
3^6=29 mod 100 3^12=41 mod 100 3^24=81 mod 100 but 3^4 = 81 mod 100 therefore 3^(20n + 4)=81 mod 100 so 3^2004=81 mod 100 now 3^10=49 mod 100 this means 3^ 2014 = 69 mod 100
what i did was this: if going from 3^4 to 3^24 there was no change in the right hand side of the congruence function then we could keep on adding 20 to the power and we will still get the same result i.e. 81 so power of 2004 also gives 81 when put through congruence function. After that we proceed manually by finding 3^10 mod 100 and hence obtain the result by multiplication of that with 81 I am loving this solution :D
{ 3 }^{ 2014 } \equiv { ({ 3 }^{ 20 }) }^{ 100 }\times { 3 }^{ 14 }\left[ 100 \right] \ \ { 3 }^{ 20 }\equiv 1\left[ 100 \right] \ { 3 }^{ 14 }\equiv 69\left[ 100 \right] \ \ alors\quad :\quad { 3 }^{ 2014 }\equiv 69\left[ 100 \right]
Phi denotes number of positive integers not exceeding that is relative prime to the integer.
3^15== 07 (mod 100) ..<..> 7^4 == 01 (mod 100)......
.==means equivalent.
3^2014 == (3^15)^134 * 3^4 (mod 100) == (7^4)^33 * 7^2 * 3^4 (mod 100)
7^2 * 3^4 (mod 100) == 69.
So, we know that φ ( 1 0 0 ) = 4 0 . This means that 3 2 0 1 4 ≡ 3 2 0 1 4 − 4 0 ≡ 3 2 0 1 4 − 2 ( 4 0 ) ≡ . . . ≡ 3 2 0 1 4 − 5 0 ( 4 0 ) ≡ 3 1 4 ( m o d 1 0 ) . This means the answer is 6 9