Final answer:
The true statement about the factors of the number with prime factorization 3² × 5³ × 7 is that the number is divisible by 105. This is because the number includes all the prime factors of 105 within its prime factorization. Other options can be dismissed based on the principles of calculating the number of factors and recognizing perfect squares. The correct answer is option B .
Step-by-step explanation:
The prime factorization of a number is given as 3² × 5³ × 7. To determine which statement about the factors of this number is true, let's break down each of the given options and apply some basic mathematical principles.
Option (b) suggests that the number is divisible by 105. Since 105's prime factorization is 3 × 5 × 7, and our number has these prime factors (and more), we can immediately confirm that the given number is indeed divisible by 105. This matches the mathematical rule of divisibility that a number is divisible by another if it contains all prime factors of the latter number (possibly with higher exponents).
Looking at option (a), that there are 180 factors is less intuitive, but the number of factors can be calculated using the formula for determining the total number of factors from prime factorization, which is: (exponent of prime 1 + 1) × (exponent of prime 2 + 1) × ... Applying this formula here, we would calculate (2+1)×(3+1)×(1+1), which yields 24, not 180. Therefore, this statement is incorrect.
The sum of the factors (option c) is more complex to compute and is not easily done without a structured method or a computer algorithm; hence, we will not be able to confirm this statement's accuracy in this context. Finally, regarding option (d), for a number to be a perfect square, all the exponents in its prime factorization must be even. Since the exponent for prime number 7 is 1, an odd number, the given number is not a perfect square.
Therefore, it is clear that the true statement about the factors is option (b), and that the number is divisible by 105.