The factors of 12
A factor of 12 is a whole number that divides 12 exactly, with nothing left over. Factors come in pairs that multiply to 12: 1 × 12, 2 × 6 and 3 × 4. So the factors of 12 are 1, 2, 3, 4, 6 and 12.
The six factors of 12 are ringed: 1, 2, 3, 4, 6 and 12.
The factors of 18
Do the same for 18. Its factor pairs are 1 × 18, 2 × 9 and 3 × 6. 4 does not divide 18, because 18 ÷ 4 = 4 remainder 2, and 5 does not either. So the factors of 18 are 1, 2, 3, 6, 9 and 18.
18 = 1 × 18: 1 divides 18, so it names its partner 18; 2 factors so far
Widen the rows from 1 until the pairs cross
Widen the rows of 18 squares one step at a time. When the width divides 18, the squares make a full rectangle, and its width and number of rows are a factor pair. At a width of 4 the last row is short, so 4 is not a factor.
Factors they share
Now compare the two lists. 1, 2, 3 and 6 are in both: each of them divides 12 and also divides 18. A number that divides both of two numbers is a common factor of them. The common factors of 12 and 18 are 1, 2, 3 and 6.
The other factors belong to one number only. 4 and 12 divide 12 but not 18. 9 and 18 divide 18 but not 12.
The factors of 12 fill the left circle and the factors of 18 the right one. The common factors, 1, 2, 3 and 6, are in the overlap, because they belong to both.
Sometimes only 1
1 divides every whole number, so 1 is a common factor of any two numbers. For some pairs it is the only one. The factors of 8 are 1, 2, 4 and 8, and the factors of 9 are 1, 3 and 9. The only number in both lists is 1.
8 and 9 share only the factor 1.
Test both numbers
A common factor must divide both numbers, so test each one against both. 4 divides 12, but 18 ÷ 4 leaves 2, so 4 is not a common factor of 12 and 18. 9 divides 18, but 12 ÷ 9 leaves 3.
A factor is never bigger than its number, so a common factor is never bigger than the smaller of the two numbers. Every common factor of 12 and 18 is 12 or less.