Question

If A=m^3×n×p^2, where m, n and p are prime numbers, which of the following is a perfect square?

Answer 1

Answer:

Theorem 4.9 in Section 4.2 states that every natural number greater than 1 is either a prime number or a product of prime numbers. When a composite number is written as a product of prime numbers, we say that we have obtained a prime factorization of that composite number. For example, since \(60 = 2^2 \cdot 3 \cdot 5\), we say that \(2^2 \cdot 3 \cdot 5\) is a prime factorization of 60.

Write the number 40 as a product of prime numbers by first writing \(40 = 2 \cdot 20\) and then factoring 20 into a product of primes. Next, write the number 40 as a product of prime numbers by first writing \(40 = 5 \cdot 8\) and then factoring 8 into a product of primes.

In Part (2), we used two different methods to obtain a prime factorization of 40. Did these methods produce the same prime factorization or different prime factorizations? Explain.

Repeat Parts (2) and (3) with 150. First, start with \(150 = 3 \cdot 50\), and then start with \(150 = 5 \cdot 30\)