Graham's GCD problem asks whether every set of 
distinct positive integers 
satisfies
 |
where 
is the greatest common divisor (Graham
1970). The maximum is taken over ordered pairs, so interchanging 
and 
can change the quotient.
Balasubramanian and Soundararajan (1996) proved the assertion for every 
. The bound is sharp for the set 
, since every quotient is at
most 
and the pair 
attains 
.
This is problem 402 in the Erdős problems collection (Bloom 2026).
