+166 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 $\gcd(b,c)$?
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.
