Yes, a pair of numbers can have fewer than two common factors. This only happens when the numbers are co-prime -- that is, when the numbers have only one common factor. In that case, the common factor is always 1.
All numbers have common multiples.
Even numbers.
20 and 40 have 4 and 5 as common factors.
Okay. Give us a pair of numbers.
The common factor is 1.
12 and 24
12 and 24
21 and 42
' 1 ' is their only common factor. They're both prime numbers.
Just 1.
20 and 40
I suggest factoring each pair of numbers, and checking whether they have, or don't have, common factors. A pair of numbers is said to be "relatively prime" if they have no common factors (their greatest common factor is 1). For larger numbers, Euclid's algorithm could be used, but for such small numbers, factoring is probably faster.