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If a, b, and c are positive integers such that gcd(a,b) = 168 and gcd(a,c) = 490, then what is the smallest possible value of gcd(b,c)?

 May 17, 2022
 #1
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a ==5,880,  b==6,048,  c==6,370

 

GCD(a, b) ==GCD(5,880, 6,048) ==168

 

GCD(a, c) ==GCD(5,880, 6,370) ==490

 

GCD(b, c) ==GCD(6,048, 6,370) ==14

 May 17, 2022
 #2
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b divides 168 and c divides 490 by definition of gcd.

\(\gcd(b, c) \geq \gcd(168, 490) = 14\)

 

Therefore the smallest value is 14.

 May 18, 2022

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