Let n be a natural number. If n is divisible by 9, that is neither a necessary nor sufficient condition for n being divisible by 6, right? Because I'm not sure about the wording of the question asking whether it is either necessary or sufficient, but I believe it is neither.
For example:
6*5 = 30
So 30 is divisible by 6, but is not divisible by 9.
Therefore, n being divisible by 9 is neither a necessary nor sufficient condition for n being divisible by 6.
clearly, since 9 is divisible by 9, but not by 6, it is not sufficient.
On the other hand, 12 is not divisible by 9, but is divisible by 6, so it is not necessary.
You are correct in your understanding. The condition of a natural number n being divisible by 9 is neither a necessary nor a sufficient condition for n being divisible by 6.
To explain further, we need to understand the concepts of necessary and sufficient conditions:
1. Necessary condition: A necessary condition is something that must be true for the statement to be true. If the necessary condition is not fulfilled, then the statement will definitely be false.
2. Sufficient condition: A sufficient condition is something that, if true, guarantees the truth of the statement. If the sufficient condition is fulfilled, then the statement will definitely be true.
In this case, the condition of n being divisible by 9 is not necessary for n to be divisible by 6 because there are numbers (such as 30 in your example) that are divisible by 6 but not by 9. Therefore, n being divisible by 9 is not a necessary condition for n being divisible by 6.
Similarly, being divisible by 9 is not a sufficient condition for being divisible by 6 because there are numbers (such as 18) that are divisible by both 6 and 9. However, this does not mean that being divisible by 9 guarantees being divisible by 6. There are numbers (such as 15) that are divisible by 9 but not by 6.
So, in summary, being divisible by 9 is neither a necessary condition nor a sufficient condition for being divisible by 6.