Analogy Explanation:

The given analogy is "Up: Down :: Factor : _____".

Understanding the Analogy:

In this analogy, we are given a pair of words: Up and Down. These two words are antonyms, meaning they have opposite meanings. Up represents the upward direction and Down represents the downward direction.

We are asked to find a word that has a similar relationship with the word "Factor". To determine the relationship, we need to understand the meaning of the word "Factor" and find a word that is related to it in a similar way as Up is related to Down.

Explanation:

- Up and Down are opposite directions - one is the opposite of the other.
- Similarly, a factor and its counterpart are also opposite in a sense. A factor is a number that divides another number without leaving a remainder. So, the counterpart of a factor would be a number that is divided by another number without leaving a remainder.

Options:

a) Multiple: A multiple of a number is a number that can be divided by that number without leaving a remainder. It is the opposite concept of a factor. Therefore, this option fits the analogy correctly.

b) Prime: A prime number is a number that is only divisible by 1 and itself. It is not directly related to the concept of factors. Therefore, this option does not fit the analogy.

c) Composite: A composite number is a number that has more than two factors. It is related to factors, but it does not represent the opposite concept. Therefore, this option does not fit the analogy.

d) Divide: While divide is the action of splitting or separating into parts, it does not represent the opposite concept of a factor. Therefore, this option does not fit the analogy.

Conclusion:

Based on the explanation and understanding of the analogy, the correct option is a) Multiple.