gcd and divisibility

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 Joined: Mon May 09, 2016 11:18 am
gcd and divisibility
If a multiple of 864 is taken which is a positive integer,then what is the probability of that number to be divisible by 1944
 nahin munkar
 Posts: 81
 Joined: Mon Aug 17, 2015 6:51 pm
 Location: banasree,dhaka
Re: gcd and divisibility
The ans is $ \dfrac{1}{ 9} $.
Last edited by nahin munkar on Mon Aug 22, 2016 1:57 pm, edited 1 time in total.
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss

 Posts: 17
 Joined: Mon May 09, 2016 11:18 am
Re: gcd and divisibility
can u describe the probability part a bit more clearly?
 asif e elahi
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 Joined: Mon Aug 05, 2013 12:36 pm
 Location: Sylhet,Bangladesh
Re: gcd and divisibility
Take $864k$ as a multiple of $864$. Now if $1944$ divides $864k$, we must have $\dfrac{864k}{1944}\in \mathbb{N}$. Which implies $9\mid k$. The probability of happening this is $\dfrac{1}{9}$ when we choose $k$ randomly.

 Posts: 17
 Joined: Mon May 09, 2016 11:18 am
Re: gcd and divisibility
I got that divisibility part that 9 is the factor of k but I I didn't get the probability part.if u could interpret how to determine the probability It would be percieved...thanx
 nahin munkar
 Posts: 81
 Joined: Mon Aug 17, 2015 6:51 pm
 Location: banasree,dhaka
Re: gcd and divisibility
OK . If $ 9k$, then $k$ is divided by $9$ only when it is a multiple of $9$. Then u will get $ k$ as a multiple of $9$ after each $9$ numbers . Observe $9,18,27,36..$...etc are multiple of $9$ .Look they r returning back as each $9th$ number ($8$ number after after). If u make use of probability formula then u get that $ k$ returns as a multiple of $9$ after each $9$ numbers. So, u'll get 1 number from any consequent $9$ numbers. Mainly, the probability version tells us u will get 1 number k as multiple of $9$ if u choose $9$ numbers.So, the probability is $ \dfrac{1}{9}$ as follows.kh ibrahim wrote:I got that divisibility part that 9 is the factor of k but I I didn't get the probability part.if u could interpret how to determine the probability It would be percieved...thanx
# Mathematicians stand on each other's shoulders. ~ Carl Friedrich Gauss