Italian MO 2002#P1

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Phlembac Adib Hasan
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Italian MO 2002#P1

Unread post by Phlembac Adib Hasan » Tue Jun 11, 2013 8:35 am

Find all $3$-digit positive integers that are $34$ times of their digital sum.

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asif e elahi
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Re: Italian MO 2002#P1

Unread post by asif e elahi » Tue Aug 27, 2013 11:29 pm

What is digital sum?

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Fatin Farhan
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Re: Italian MO 2002#P1

Unread post by Fatin Farhan » Wed Aug 28, 2013 10:47 am

$34*3*n$
such that n is an integer and $0<n<5$
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

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Fatin Farhan
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Re: Italian MO 2002#P1

Unread post by Fatin Farhan » Wed Aug 28, 2013 11:38 am

asif e elahi wrote:What is digital sum?
Digit sum is the sum of digits.
Let $D$ be a three digit number where D=100a+10b+c. Then digit sum of D is $a+b+c$
"The box said 'Requires Windows XP or better'. So I installed L$$i$$nux...:p"

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Fahim Shahriar
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Re: Italian MO 2002#P1

Unread post by Fahim Shahriar » Wed Aug 28, 2013 8:36 pm

$34 \times (a+b+c) = 100a+10b+c$
$\Rightarrow 66a-24b-33c = 0$
$\Rightarrow 11 \times (2a-c) = 8b$

$8b$ will be divisible by $11$ if $b=0$.

Now, $11 \times (2a-c)=0$
$\Rightarrow 2a = c$

So the numbers are $102,204,306,408$.
Name: Fahim Shahriar Shakkhor
Notre Dame College

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Mursalin
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Re: Italian MO 2002#P1

Unread post by Mursalin » Wed Aug 28, 2013 8:38 pm

@Fahim

Nicely done!
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