Problem 7 - Intermediate Algebra

Question

In the following exercises, find the prime factorization. 86

Step-by-Step Solution

Answer

The prime factors of 86 are 2 and 43.

Step 1: Identify if the number is even or odd

Check if 86 is divisible by 2. Since 86 is an even number, it is divisible by 2.

Step 2: Divide by 2

Divide 86 by 2 to get the quotient:86 ÷ 2 = 43.

Step 3: Check if 43 is a prime number

Determine whether 43 can be divided evenly by any prime number smaller than its square root (~6.6). Since 43 is not divisible by 2, 3, or 5, it is a prime number.

Step 4: Combine the prime factors

Since 43 is a prime number and the only other prime factor is 2, the prime factorization of 86 is 2 and 43.

Key Concepts

Prime Numbers

Prime numbers are essential in math. A prime number has no divisors other than 1 and itself. For instance, 2 is the smallest prime number. It's only divisible by 1 and 2. Other examples include 3, 5, 7, 11, and so on. These numbers form the building blocks for other numbers in mathematics. This concept is critical when discussing prime factorization. Prime factorization means breaking down a number into its prime factors. Knowing prime numbers helps in identifying these factors quickly and accurately.

Divisibility Rules

Divisibility rules are shortcuts that help determine if one number divides another without a remainder. Understanding these rules simplifies the factoring process.
- **Divisibility by 2:** A number is divisible by 2 if it's even. In other words, if it ends in 0, 2, 4, 6, or 8.
- **Divisibility by 3:** If the sum of a number's digits is divisible by 3, then the number itself is too. For example, 123 is divisible by 3 because 1 + 2 + 3 = 6, and 6 is divisible by 3.
- **Divisibility by 5:** Any number ending in 0 or 5 is divisible by 5. Divisibility rules make finding factors faster and more straightforward, especially when verifying if a number is prime. Using these rules, we can quickly identify that 86 is divisible by 2, leading us to its prime factorization efficiently.

Factoring Process

Factoring involves breaking down a number into its component parts, or factors. To factor 86, we follow these steps:
- **Step 1:** Determine if 86 is even or odd. Since it's even, we know it's divisible by 2. This knowledge aligns with the divisibility rules. Step 1 simplifies the process.
- **Step 2:** Divide by 2. Here, we get 86 ÷ 2, which equals 43. The number 43 now needs to be examined.
- **Step 3:** Check if 43 is prime. We do this by testing divisibility by all prime numbers smaller than its square root (approximately 6.6). Since 43 isn't divisible by 2, 3, or 5, we confirm it's prime.
The factors of 86 are thus found to be 2 and 43. This systematic approach ensures accuracy and makes complex problems manageable.