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#### mathmaniac

##### Active member

- Mar 4, 2013

- 188

Prove that there exists a power of 3 that ends in 001.

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- Mar 4, 2013

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Prove that there exists a power of 3 that ends in 001.

- Mar 10, 2012

- 835

consider the integers:Prove that there exists a power of 3 that ends in 001.

$3^1,3^2,\ldots,3^{1001}$

Some two distinct integers of these must leave the same remainder mod $1000$.

So there exist distinct $i$ and $j$ such that $3^i\equiv 3^j\pmod{1000}$.

WLOG $i<j$. Thus $3^i(3^{j-i}-1)\equiv 0\pmod{1000}$

Thus $3^{j-1}\equiv 1\pmod{1000}$ and we are done.

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- Mar 4, 2013

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Good work!!!So quick!!!

Now what odd numbers can we replace for 001???

Now what odd numbers can we replace for 001???

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- Mar 10, 2012

- 835

I don't know if there's an analytic way to enumerate all such numbers. But it can be shown then the number of such numbers which are less then 1000 divides $\phi(1000)$.Good work!!!So quick!!!

Now what odd numbers can we replace for 001???