Find the number of x

For discussing Olympiad Level Number Theory problems
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Avik Roy
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Find the number of x

Unread post by Avik Roy » Thu Dec 09, 2010 6:39 pm

Find the number of solution of the equation $2^x=4^y=8^z$ where $x$, $y$ and $z$ are positive integers, $x$ is less than $10000$ and for every $x$ satisfying the equation, sum of digits of $x$ also belongs to the solution set of $x$
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Zzzz
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Re: Find the number of x

Unread post by Zzzz » Fri Dec 10, 2010 11:50 am

Here is the solution:
$2^x=4^y=2^{2y} \Rightarrow x=2y$ implies that $2|x$.
$2^x=8^z=2^{3z} \Rightarrow x=3z$ implies that $3|x$.
$\therefore 6|x$
There are $\lfloor \frac{10000}{6}\rfloor = 1666$ solutions for $x$ if we don't consider the sum of digits.
Now, highest possible sum of digit is $9+9+9+9=36$
As, $x$ is divisible by $3$, the sum of digits of $x$ is also divisible by $3$.
Again $9999$ isn't a solution for $x$. The next possible 'sum of digits' is $33$.
We claim that for any number $k$ where $k\le33$ and $3|k$, $k$ belongs to the solution set.
Lets prove it!
We can write $k=m+n$ [here, $n \in \{2,4,6,8\}$]
Thus, we can easily keep the value of $m$ under $27$.
So we can write $m=m_1+m_2+m_3$ Here $0\le m_1,m_2,m_3 \le 9$
Now, consider the number $j=\overline{m_1m_2m_3n}$. It is an even number ($n$ even) and it is divisible by $3$ (Since the sum of its digits =$k$, which is divisible by $3$).
$\therefore 6|j$
So, $j$ belongs to solution set and so does $k$.
There are $\frac{33}{3}=11$ possible values for $k$ but we have already counted $\lfloor \frac {33}{6}\rfloor=5$ values while counting without considering the sum of digits.
So, the answer is \[1666+11-5\] \[=1672\]
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Avik Roy
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Re: Find the number of x

Unread post by Avik Roy » Fri Dec 10, 2010 8:12 pm

The answer is incorrect. The correct answer is (as we include all poitive, negative integers and zero) $1665$
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Zzzz
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Re: Find the number of x

Unread post by Zzzz » Fri Dec 10, 2010 8:21 pm

O O... Probably I haven't understood the question.. Does $12$ belong to the solution set?
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Avik Roy
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Re: Find the number of x

Unread post by Avik Roy » Fri Dec 10, 2010 8:29 pm

Yes, the question should be understood as "if $a$ is a valid value of $x$ then the sum of digits of $a$ also has to be a valid value of $x$"
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rakeen
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Re: Find the number of x

Unread post by rakeen » Sat Dec 11, 2010 5:06 pm

Can u post me some properties of Prime.(especially for BdMO)
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Avik Roy
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Re: Find the number of x

Unread post by Avik Roy » Sat Dec 11, 2010 6:16 pm

rakeen wrote:Can u post me some properties of Prime.(especially for BdMO)
If you post a different topic regarding this, that will be better. This topic is discussing a particular problem and you should post replies regarding this problem only.
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Masum
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Re: Find the number of x

Unread post by Masum » Sat Dec 11, 2010 8:16 pm

One one thing is neutral in the universe, that is $0$.

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rakeen
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Re: Find the number of x

Unread post by rakeen » Mon Dec 13, 2010 4:03 pm

hope I'll have to read TnC for the first time. @Av..
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