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What will be the ones digit of the remainder in: $$\frac{\left|5555^{2222} + 2222^{5555}\right|} {7}$$

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What have you tried? What do you know? Does Modulo Arithmetic sound familiar? How about Euler's Theorem? –  Calvin Lin Jan 15 '13 at 16:01
    
I tried it like 5555...will occur 2222 times, and 5555/7 gives remainder = 4. So 4 will occur 555 times. That's what i thought and similar way for 2222. –  Ravi Jan 15 '13 at 16:04
    
It's somewhat confusing - the remainder when dividing by $7$ is already always a one-digit number, so do you mean the ones-digit of the quotient, or do you really mean the ones-digit of the remainder? Specifically, if someone asked "what is the one's digit of the remainder in $$\frac{99}{7}$$ what do you think the answer is supposed to be? –  Thomas Andrews Jan 15 '13 at 16:23
    
Its the one's digit of remainder only. I ain't getting how to procede in such a question.. –  Ravi Jan 15 '13 at 16:24
    
Well the answer is 0. –  Ravi Jan 15 '13 at 16:25

2 Answers 2

up vote 7 down vote accepted

Hint: $ 5555 \equiv 4 \pmod{7}$, $2222 \equiv 3 \pmod{7}$.

Edit: Calculate that $4^3 \equiv 1, 3^6 \equiv 1 \pmod{7}$. Hence, this implies that $4^{3k} \equiv (4^3)^k \equiv 1^k \equiv 1 \pmod{7}, 3^{6j} \equiv (3^6)^j \equiv 1^j \equiv \pmod{7}$.

Now, $2222 \equiv 2 \pmod{3}$, and $5555 \equiv 5 \pmod{6}$. Hence,

$$ 5555^{2222} + 2222^{5555} \equiv 4^{2222} + 3^{5555} \equiv 4^ 2 + 3^ 5 \pmod{7}.$$

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Hmm..the answer is 0 but by doing 269/7 it comes out 2. –  Ravi Jan 15 '13 at 16:08
    
@Ravi Please check your arithmetic again. –  Calvin Lin Jan 15 '13 at 16:10
    
Actually when you divide 5555 by 7 then 4 occurs 557 times, and when you divide 2222 by 7 then 3 occurs 1853 times. But that doesn't help either. –  Ravi Jan 15 '13 at 16:17
    
And i don't think we have to apply Euler 's theorem here. By the way , if we ignore the mod function then can you tell how to procede? –  Ravi Jan 15 '13 at 16:18
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@Ravi When I said to check your arithmetic, I meant $4^2 + 3^5 \neq 269$. You'd have to apply something to reduce the powers from 2222/5555 to some much more manageable. It need not be Euler's theorem, but that's a fast guaranteed approach. You may look at $2^3, 5^3 \pmod{7}$ and decide that's sufficient. –  Calvin Lin Jan 15 '13 at 16:26

This is not a solution. I was trying to find out the possible values of $n$ such that $${\underbrace{22\cdots22}_{n\text{ digits }}}^{\underbrace{55\cdots55}_{n\text{ digits }}}+{\underbrace{55\cdots55}_{n\text{ digits }}}^{\underbrace{22\cdots22}_{n\text{ digits }}}$$ is divisible by $7$.

Let $$\underbrace{11\cdots11}_{n\text{ digits }} =\frac{10^n-1}9=M$$

Clearly $M$ is odd.

Now, $$(2M)^{5M}+(5M)^{2M}\equiv (2M)^{5M}+(-2M)^{2M}\pmod 7\equiv (2M)^{5M}+(2M)^{2M}=(2M)^{2M}\{(2M)^{3M}+1\}$$ which will be divisible by $7$

if (i) $7\mid \{(2M)^{3M}+1\}\iff 7\mid \{M^{3M}+1\}$ as $2^3\equiv1$

$\iff 7\mid \{\left(\frac{10^n-1}9\right)^{3M}+1\}\iff 7\mid \{(10^n-1)^{3M}+1\}$ as $9^3=3^6\equiv1\pmod 7$

So, we need $(10^n-1)^{3M}\equiv-1\pmod 7\implies (10^n-1)^3\equiv-1$ as $3M\equiv3\pmod {\phi(7)}$ as $M$ is odd.

Taking Discrete Logarithm wrt a primitive root $3\mod 7,$

$3\cdot ind_3(10^n-1)\equiv 3\pmod 6=6c+3$ for some integer $c$

So, $$ ind_3(10^n-1)=2c+1$$

So, $10^n-1\equiv 3^{2c+1}\pmod 7\equiv 3,6,5\iff 10^n\equiv4,6,7\pmod7$

$10^n\not\equiv7\pmod 7$ as $(10,7)=1$

$10^1\equiv3,10^2\equiv3^3\equiv2,10^3\equiv2\cdot10\equiv6,10^4\equiv2^2=4,10^5\equiv2\cdot6\equiv5,10^6\equiv 6^2\equiv1\pmod 7$

So, $n\equiv3,4\pmod 6$

or if (ii)$7\mid (2M)^{2M}\iff 7\mid M^{2M}\implies 7\mid M\implies 10^n\equiv1\pmod 7\iff 6\mid n$

So, if $n\equiv0,3,4\pmod 6,$ $${\underbrace{22\cdots22}_{n\text{ digits }}}^{\underbrace{55\cdots55}_{n\text{ digits }}}+{\underbrace{55\cdots55}_{n\text{ digits }}}^{\underbrace{22\cdots22}_{n\text{ digits }}}$$ is divisible by $7$.

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This is not a comment. This is just a tribute. –  im so confused Jan 15 '13 at 20:22

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