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If $a$, $b$, and $c$ are positive integers such that $\gcd(a,b) = 168$ and $\gcd(a,c) = 693$, then what is the smallest possible value of $\gcd(b,c)$?

 Apr 26, 2018
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gcd(a,b)=168- this means that a=a1*168, b=b1*168, where a1 and b1 are coprime.

 

gcd(a,c)=693, meaning that a=a2*693, c=c1*693, where a2 and c1 are coprime.

 

168=23*3*7, 693=32*7*11 gcd(168, 693)=3*7=21. Therefore gcd(b,c)>=21. To prove that 21 is the smallest possible value of gcd(b,c) i have to find an example-

 

a=lcm(168, 693)=5544, b=168, c=693.

 

check:

 

gcd(a,b)=gcd(5544,168)=gcd(168*33, 168)=168*gcd(33,1)=168

 

gcd(a,c)=gcd(5544,693)=gcd(693*8,693)=693*gcd(8,1)=693

 

gcd(b,c)=gcd(168,693)=21.

 Apr 26, 2018
edited by Guest  Apr 26, 2018

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