If ‘p’ is a multiple of 7 and p = a^3 * b

This is a ‘must be’ question. Therefore, we will have to test out multiple cases and the option that is true in all cases will be the answer.

The strategy for ‘must be’ type of questions, is to try and falsify the options given. If an option is false even once, it means that it is not true in all cases and hence you cannot say that the option/statement must be true. Taking simple cases (values), you will be able to eliminate at least 2 out of the 5 options. Eliminating the 3rd and the 4th option may entail taking up few more smart cases; the option left out HAS to be the correct answer. This is the strategy to be followed in all ‘must be’ questions.

Let us now look at the question under discussion.
We know that p = \(a^3 * b\) and also that p is a multiple of 7.
Therefore, \(a^3\) * b = a multiple of 7. This means that at least one of a or b should be a multiple of 7.

Case 1: ‘a’ is a multiple of 7 and ‘b’ is not.

If a is a multiple of 7, \(a^3\) will be a multiple of 343. However, since b is not a multiple of 7, \(b^3\) is not a multiple of 343. Therefore, the options that contain \(b^3\) can be eliminated, since these options will give a false case. So, options B and C can be eliminated.

Case 2: ‘b’ is a multiple of 7 and ‘a’ is not.

On similar lines, options containing \(a^3\) can now be eliminated since these will not be a multiple of 343 necessarily. So, options A and E can be eliminated.

The only option left is option D. This has to be the answer. But instead of just concluding it like this, let us also look at analyzing the option.

Option D talks about \(a^3*b^3\). This can also be written as \((ab)^3\). Regardless of whether ‘a’ is a multiple of 7 or ‘b’, ‘ab’ will definitely be a multiple of 7. If ‘ab’ is a multiple of 7, \((ab)^3\) will certainly be a multiple of \(7^3\) i.e. 343.

So, the correct answer option is D.

As mentioned earlier, the best way to tackle ‘must be’ questions is to take simple cases and eliminate options, by falsifying them once. The option that is left at the end of this process will be the answer.

Hope this helps!