BdMO National Higher Secondary 2007/14

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BdMO
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BdMO National Higher Secondary 2007/14

Unread post by BdMO » Sun Feb 06, 2011 10:23 pm

Problem 14:
If $m +12 = p^a$ and $m -12 = p^b$ where $a,b,m$ are integers and $p$ is a prime number. Find all possible primes $p > 0$ . [Note: $p$ only takes three values]

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Moon
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Re: BdMO National Higher Secondary 2007/14

Unread post by Moon » Sun Feb 06, 2011 10:30 pm

This is our $500^{th}$ topic. :D
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Tahmid Hasan
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Re: BdMO National Higher Secondary 2007/14

Unread post by Tahmid Hasan » Sun Feb 06, 2011 10:37 pm

let's name the eqs $(i)$ and $(ii)$ and do $(i)-(ii)$
than divide three cases
1.$a>b$
2.$a<b$
3.$a=b$
বড় ভালবাসি তোমায়,মা

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Cryptic.shohag
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Re: BdMO National Higher Secondary 2007/14

Unread post by Cryptic.shohag » Tue Feb 08, 2011 12:29 pm

Tahmid Hasan wrote:let's name the eqs $(i)$ and $(ii)$ and do $(i)-(ii)$
than divide three cases
1.$a>b$
2.$a<b$
3.$a=b$
Actually it's not necessary as it's clear that $a>b$.

$m+12=p^a$....(i)
$m-12=p^b$....(ii)

So, $p^a-p^b=24$. As $p^a$ and $p^b$ both are multiples of p, we can write $p^a-p^b=kp$ where k is an integer. So,
$kp=24$
As k is an integer, so p has to be a divisor of 24. Among the divisors of 24 we find 2 prime numbers and they are 2 and 3. So, p=2,3....
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein

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Cryptic.shohag
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Re: BdMO National Higher Secondary 2007/14

Unread post by Cryptic.shohag » Tue Feb 08, 2011 12:54 pm

Oops!! I missed one solution. In last solution I considered only $a,b>0$....
As $a>b$ and $p^a-p^b=24$, a's value cannot be 0. So, if we consider $b=0$, then we get,
$p^a-1=24$
$p^a=25$
$p^a=5^2$....
So, another value of p is 5....
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein

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Cryptic.shohag
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Re: BdMO National Higher Secondary 2007/14

Unread post by Cryptic.shohag » Tue Feb 08, 2011 12:55 pm

So, p=2,3,5.... :)
God does not care about our mathematical difficulties; He integrates empirically. ~Albert Einstein

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