If the least common multiple of two 6-digit integers has 10 digits, then their greatest common divisor has at most how many digits?

er1004 Apr 25, 2019

#2**0 **

Here is my attempt:

The maximum LCM of two 6-digit numbers is 2 x 6=12 digits. Example:993053 x 993043=990,116,542,279(12 digits)

The minimum LCM of two 6-digit numbers is 2 x 6 -1=11 digits. Example:100213x100847=10,106,180,411(11 digits)

If their LCM is 10 digits, then that means they must share one common divisor between them. That common divisor could be a 1-digit number or a 2-digit number at a maximum.

Example:LCM{110210, 100500} = 1,107,610,500(10 digits)

GCD {110210, 100500} =10 (2 digits)

**Therefore, the GCD of the two 6-digit numbers can have at most 2 digits.**

Guest Apr 25, 2019