Final answer:
The proof lies in the fact that if a 3-digit composite number had no prime factors less than or equal to 31, it would be a prime number. However, since the square root of the largest 3-digit number (999) is less than 32, a composite must have a prime factor of 31 or less.
Step-by-step explanation:
The question asks for a proof that any composite 3-digit number must have a prime factor less than or equal to 31. To understand this, we can utilize the fact that if a composite number, which is greater than 1 and not a prime itself, does not have any prime factors less than or equal to its square root, then it must be a prime number.
Since the largest 3-digit number is 999, the square root of 999 is less than 32. Therefore, if a 3-digit number were to be composite, it must have a prime factor that is less than or equal to the square root of 999, which means it must have a prime factor less than or equal to 31.