## Dhaka Secondary 2011/2 (Junior 2011/4)

Problem for Secondary Group from Divisional Mathematical Olympiad will be solved here.
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BdMO
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### Dhaka Secondary 2011/2 (Junior 2011/4)

$A$ is the product of seven odd prime numbers. $A \times B$ is a perfect even square. What is the minimum number of prime factors of $B$?

Hasib
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### Re: Dhaka Secondary 2011/2

are the seven odd prime numbers distinct?
A man is not finished when he's defeated, he's finished when he quits.

Avik Roy
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### Re: Dhaka Secondary 2011/2

yes, they are distinct
"Je le vois, mais je ne le crois pas!" - Georg Ferdinand Ludwig Philipp Cantor

Hasib
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### Re: Dhaka Secondary 2011/2

then the ans is 8.
Let, $A=p_1.p_2...p_7$ none of them are same or 2. So $B=2.2.p_1.p_2...p_7$ then the ans is 8 A man is not finished when he's defeated, he's finished when he quits.

Shifat
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### Re: Dhaka Secondary 2011/2 (Junior 2011/4)

bro i did not get your solution, can u describe it even more clearly??@hasib bro

amlansaha
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### Re: Dhaka Secondary 2011/2 (Junior 2011/4)

shifat, i am giving an example. suppose $A= 3\times 5\times 7\times 11\times 13\times 17\times 19$ and as $A\times B$ is an even perfect square, it should look like this $A\times B= 2^2\times 3^2\times 5^2\times 7^2\times 11^2\times 13^2\times 17^2\times 19^2\times$(any other perfect square) . thus we get $B=2^2\times 3\times 5\times 7\times 11\times 13\times 17\times 19\times$(any other perfect square). so the minimum prime factors of $B$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$. so the answer is $8$. Hasib has just used $p_{1}$, $p_{2}$, $p_{3}$..... in lieu of $3$, $5$, $7$.... that's it অম্লান সাহা

Shifat
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### Re: Dhaka Secondary 2011/2 (Junior 2011/4)

oh, got it, thanks..... Naheed
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### Re: Dhaka Secondary 2011/2 (Junior 2011/4)

What's a perfect even square?