The statement is If a number is divisible by 2 and by 3, then it is divisible by 6 and vice versa. Is this statement true? Give an example or counterexample.

Question

The statement is If a number is divisible by 2 and by 3, then it is divisible by 6 and vice versa. Is this statement true? Give an example or counterexample.

I believe it is true but am stuck on the counterexample/example.

Answer 1

True

e.g.

24 = 2x3x4, shows it is divisible by 2 and by 3
= 6x4 shows it is divisible by 6

There is no counterexample, since the statement is always true.

Answer 2

To determine whether the statement "If a number is divisible by 2 and by 3, then it is divisible by 6 and vice versa" is true or not, we can start by considering the definition of divisibility.

For a number to be divisible by another number, it means that when the first number is divided by the second number, there is no remainder. In other words, the division is exact.

Let's consider the case where a number is divisible by both 2 and 3. This means that when we divide the number by 2 and 3, respectively, there are no remainders.

If a number is divisible by 2, it means it is an even number since even numbers are exactly divisible by 2.

If a number is divisible by 3, it means that the sum of its digits is divisible by 3.

Now, let's consider the case where a number is divisible by 6. A number is divisible by 6 if it is both divisible by 2 (an even number) and by 3 (the sum of its digits is divisible by 3).

So, if a number is divisible by 2 and 3, we can conclude that it is divisible by 6.

Now let's consider the vice versa part of the statement, which states that if a number is divisible by 6, then it is divisible by both 2 and 3.

Suppose we have a number such as 12. 12 is divisible by 6 since 12 divided by 6 gives us 2 with no remainder. Additionally, 12 is also divisible by 2 since it is an even number. However, 12 is not divisible by 3 since the sum of its digits (1+2) is not divisible by 3. Therefore, this example serves as a counterexample to the statement.

In conclusion, the statement "If a number is divisible by 2 and by 3, then it is divisible by 6 and vice versa" is not entirely true, as we have demonstrated with the counterexample of the number 12.