MHB Find Last Two Digits of ${2017}^{82}$

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To find the last two digits of 2017 raised to the power of 82, the calculation involves determining 2017 mod 100, which simplifies to 17. Using Euler's theorem, since 2017 is coprime to 100, we calculate Euler's totient function φ(100) to be 40. This leads to the conclusion that 2017^40 ≡ 1 mod 100, allowing us to reduce 2017^82 to 2017^2 mod 100. Ultimately, 2017^2 mod 100 equals 89, providing the last two digits of 2017^82 as 89.
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What are the last two digits of ${2017}^{82}$? We have only learned Fermat's little theorem and Euler's generalization so I am not sure how they apply?
 
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Deanmark said:
What are the last two digits of ${2017}^{82}$? We have only learned Fermat's little theorem and Euler's generalization so I am not sure how they apply?

Let's try to apply Euler's generalization.
To find the last two digits we need ${2017}^{82} \bmod 100$, so we have $n=100$.
How far can you get calculating Euler's totient function $\phi(100)$?
 
I like Serena said:
Let's try to apply Euler's generalization.
To find the last two digits we need ${2017}^{82} \bmod 100$, so we have $n=100$.
How far can you get calculating Euler's totient function $\phi(100)$?

I don't know anything about it. We only went of Fermat's and Euler's generalization of ax$\equiv$ b mod(m). The class is a Ring Theory class so we aren't really diving that deeply into the number theory.
 
Deanmark said:
I don't know anything about it. We only went of Fermat's and Euler's generalization of ax$\equiv$ b mod(m). The class is a Ring Theory class so we aren't really diving that deeply into the number theory.
$\displaystyle \varphi(n) :=n \prod_{p\mid n} \left(1-\frac{1}{p}\right)$ and $100 = 2^25^2$ so $\displaystyle \varphi(100) =100 \prod_{p\mid 100} \left(1-\frac{1}{p}\right) = 100 \left(1-\frac{1}{2}\right) \left(1-\frac{1}{5}\right) = 40. $

Euler's Theorem: if $a$ and $n$ are coprime, then $\displaystyle a^{\varphi(n)} \equiv 1 \bmod{n}$ where $\varphi(n)$ is given by the above product.

Since $2017$ is relatively coprime to $100$, what does that tell you?
 
June29 said:
$\displaystyle \varphi(n) :=n \prod_{p\mid n} \left(1-\frac{1}{p}\right)$ and $100 = 2^25^2$ so $\displaystyle \varphi(100) =100 \prod_{p\mid 100} \left(1-\frac{1}{p}\right) = 100 \left(1-\frac{1}{2}\right) \left(1-\frac{1}{5}\right) = 40. $

Euler's Theorem: if $a$ and $n$ are coprime, then $\displaystyle a^{\varphi(n)} \equiv 1 \bmod{n}$ where $\varphi(n)$ is given by the above product.

Since $2017$ is relatively coprime to $100$, what does that tell you?

$2017^{40} \equiv$ 1 mod (100)

$2017^{82} = 2017^{80} \cdot 2017^{2}$

$ 2017^{2} - 1 \equiv 0 \ mod (100)$

88??
 
Deanmark said:
$ 2017^{2} - 1 \equiv 0 \ mod (100)$
Could you explain what's happening here?
 
June29 said:
Could you explain what's happening here?

A mistake.

It's just $2017^{2}$ mod(100) which you can find with a calculator, but if you did not have a calculator could we transform $2017^{2}$ mod(100) further.
 
Deanmark said:
A mistake.

It's just $2017^{2}$ mod(100) which you can find with a calculator, but if you did not have a calculator could we transform $2017^{2}$ mod(100) further.
We have $2017^{2}\bmod{100} \equiv (17)^2\bmod{100} \equiv 289 \bmod{100} \equiv 89 \bmod{100}. $

By definition if $a=b+km$ for $k \in \mathbb{Z}$ then $\displaystyle a \equiv b \bmod{m}$ (first and last step).
 
Last edited:
June29 said:
We have $2017^{2}\bmod{100} \equiv (-17)^2\bmod{100} \equiv 289 \bmod{100} \equiv 89 \bmod{100}. $

By definition if $a=b+km$ for $k \in \mathbb{Z}$ then $\displaystyle a \equiv b \bmod{m}$ (first and last step).

Gotcha. I appreciate it.
 
  • #10
June29 said:
We have $2017^{2}\bmod{100} \equiv (-17)^2\bmod{100} \equiv 289 \bmod{100} \equiv 89 \bmod{100}. $

By definition if $a=b+km$ for $k \in \mathbb{Z}$ then $\displaystyle a \equiv b \bmod{m}$ (first and last step).

No

We have $2017^{2}\bmod{100} \equiv (17)^2\bmod{100} \equiv 289 \bmod{100} \equiv 89 \bmod{100}. $and further last term 89 shall do
 
  • #11
kaliprasad said:
No

We have $2017^{2}\bmod{100} \equiv (17)^2\bmod{100} \equiv 289 \bmod{100} \equiv 89 \bmod{100}. $and further last term 89 shall do
Help me understand the objection here? 'Cause $17$ and $-17$ differ by $100$.

Is there some convention you're using? The 'further last term' is obvious.
 
  • #12
June29 said:
Help me understand the objection here? 'Cause $17$ and $-17$ differ by $100$.

Nope. They differ by 34 (or 66 if we calculate the difference the other way round).
We do have $-17 \equiv 100 - 17 \equiv 83 \pmod{100}$.
 
  • #13
I like Serena said:
Nope. They differ by 34 (or 66 if we calculate the difference the other way round).
We do have $-17 \equiv 100 - 17 \equiv 83 \pmod{100}$.
Oh, of course. It should be $\mathbf{+}17$. I've edited it now.
 

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