1. a) What are the prime factors of 56 ? b) What are the composite factors of 56 ? c) What are the square number factors of 56 ?

Sure, let's break down each part of your question regarding the number **56**. --- ### **1. a) Prime Factors of 56** **Prime factors** are the prime numbers that multiply together to give the original number. To find the prime factors of 56: 1. **Start with the smallest prime number, 2:** - 56 ÷ 2 = 28 2. **Continue dividing by 2:** - 28 ÷ 2 = 14 - 14 ÷ 2 = 7 3. **Now, 7 is a prime number and cannot be divided further.** So, the prime factorization of 56 is: \[ 56 = 2 \times 2 \times 2 \times 7 = 2^3 \times 7 \] **Prime Factors of 56:** **2 and 7** --- ### **1. b) Composite Factors of 56** **Composite factors** are factors of a number that are **composite numbers** (i.e., they have more than two positive divisors). First, let's list all the positive factors of 56: \[ \text{Factors of 56} = \{1, 2, 4, 7, 8, 14, 28, 56\} \] From these, we exclude: - **1**, because it's neither prime nor composite. - **Prime numbers**, which are **2 and 7**. **Composite Factors of 56:** **4, 8, 14, 28, and 56** --- ### **1. c) Square Number Factors of 56** **Square number factors** are factors of 56 that are perfect squares (i.e., numbers like \(1 = 1^2\), \(4 = 2^2\), \(9 = 3^2\), etc.). From the list of all factors: \[ \{1, 2, 4, 7, 8, 14, 28, 56\} \] Let's identify which of these are perfect squares: - **1** is \(1^2\) - **4** is \(2^2\) **Square Number Factors of 56:** **1 and 4** --- **Summary:** - **a) Prime Factors:** 2 and 7 - **b) Composite Factors:** 4, 8, 14, 28, and 56 - **c) Square Number Factors:** 1 and 4