No, a composite number cannot have exactly three factors. By definition, a composite number is an integer greater than one that is not prime; meaning it has at least one positive divisor other than one and itself.
To understand why a composite number cannot have exactly three factors, let’s consider the properties of factors. If a number has three factors, it must be structured in a specific way. The only scenario in which a number can have three factors is when it is a perfect square of a prime number.
For example, take the number 9, which is a composite number. Its factors are 1, 3, and 9. We can see it has three factors. However, 9 is actually a square of the prime number 3 (since 3 x 3 = 9). Thus, in general, any number with exactly three factors must be of the form p², where p is a prime number. This means that while such a number is composite when constructed this way (as it is not a prime but rather the square of it), it also strictly follows this pattern and does not deviate.
For composite numbers that are not perfect squares of primes, they will have four or more factors. Hence, while you may find instances like 9 that fit the criteria of having three factors, they inherently follow a specific rule that involves primes.