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 ?

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UpStudy AI · Accepted Answer

Sure, let's break down each part of your question regarding the number **56**.

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### **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**

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### **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**

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### **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**

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**Summary:**
- **a) Prime Factors:** 2 and 7
- **b) Composite Factors:** 4, 8, 14, 28, and 56
- **c) Square Number Factors:** 1 and 4
**Prime Factors of 56:** 2 and 7 **Composite Factors of 56:** 4, 8, 14, 28, 56 **Square Number Factors of 56:** 1 and 4

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 ?

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