It is true that if 2 | 36 and 9 | 36, then 18 | 36, since 18 = 2 x 9.

Question

So wouldn’t it also be true that since 4 | 36 and 6 | 36, then 24 | 36 since 24 = 4 x 6? Why or why not?

Answer 1

2 divides 36, 9 divides 36, and therefore 2*9=18 divides 36 because 2 and 9 have no common factors (except 1)

4 divides 36, 6 divides 36, but 4*6 does not divide 36 because 4 and 6 have the common factor 2

(4*6)/2 divides 36 because we took out the common factor of 2

5 divides 75, 3 divides 75, so 5*3 divides 75 because 5 and 3 have no common factors

15 divides 75, 25 divides 75, but 15*25=375 does not divide 75

take out the (greatest) common factor of 5 and (15*25)/5=75 divides 75

This reasoning is just the reverse of the following...

You know the rules for divisibility by 2,3,4,5,6,7,8,9,10, and 11 !

Here are a few more...

A number is divisible by 12 if it is divisible by both 3 and 4 (no common factors)

A number is divisible by 15 if it is divisible by both 3 and 5 (no common factors)

A number is divisible by 18 if it is divisible by both 2 and 9 (no common factors)

A number is not divisible by 18 if it is divisible by both 3 and 6 (common factor of 3 !!!)

A number is divisible by 21 if it is divisible by both 3 and 7 (no common factors)

Look at the rule for divisibility by 6: any number divisible by both 2 and 3 is divisible by 6

A number is divisible by 22 if it is divisible by both 2 and 11

26 is 2 and 13 but the rule for divisibility by 13 isn't usually taught because of its difficulty

28 is 4 and 7

30 is 3 and 10 or 6 and 5

33 is 3 and 11

and you can make up many more rules...