Why you can't have a number in the 20 times tables and not in the 10 times tables?

Answer 1

The short answer is because 20 is itself in the 10 times table.

The long answer, which constitutes a mathematically sound proof is as follows:

Suppose x is a number in the 20 times table. That is to say that x is some integer times 20. Suppose that integer is y.

Then what we have is x = 20*y

But 20 = 10*2 so, writing 10*2 for 20 gives you x = (10*2)*y

Then by the associate property of multiplication, x = 10*(2*y)

Now, if y is an integer then 2*y is an integer. Call that integer z

So we have x = 10*z ie x is a multiple of 10.