Problem 77 - Algebra 1: Concepts and Skills

Question

List all the factors of the number. 18

Step-by-Step Solution

Answer

The factors of 18 are: 1, 2, 3, 6, 9, 18.

Step 1 - Understand the terms

First, let's understand what 'factors' mean. Factors are numbers that divide another number exactly without leaving any remainder.

Step 2 - Find the factors of 18

Start by dividing 18 by numbers starting from 1. Make sure to only consider numbers that divide exactly into 18. If the division doesn't have a remainder, then the number is a factor of 18. Repeat this process until you reach 18.

Step 3 - List down the factors

Performing the divisions, we find that the factors of 18 are 1, 2, 3, 6, 9 and 18.

Key Concepts

Divisibility

Understanding divisibility is key to recognizing the factors of a number. A number is said to be divisible by another when it can be divided without any remainder. The concept of divisibility helps us in figuring out which numbers can be considered as factors.

• Checking Divisibility: For example, when checking if 18 is divisible by 2, we perform the division 18 ÷ 2 = 9. The result is a whole number, indicating 18 is divisible by 2. Therefore, 2 is a factor.
• Each factor pair you find (like 2 and 9) shows divisibility and reveals more about the structure of the original number.
• It’s not just limited to small numbers; it remains key for larger ones too, helping streamline how we list factors.

Using divisibility rules, like the ones for 2 (even numbers) or 3 (sum of digits is divisible by 3), can simplify this search further, making factor discovery quicker and easier.

Prime Factorization

Prime factorization involves expressing any number as a product of prime numbers. It's a foundation of mathematics, revealing the building blocks of numbers. Each number has a unique prime factorization.

• Breaking it Down: To prime factorize 18, start by the smallest prime, which is 2. Check divisibility: 18 ÷ 2 = 9. Now factor 9 using 3: 9 ÷ 3 = 3. We are left with 3, which is prime.
• Hence, the prime factorization of 18 is 2 × 3 × 3, or 2 × 3².
• This technique aids in identifying all factors. Any combination of these prime factors results in a factor—for instance, combining 2 and one 3 gives us 6, another factor of 18.

Understanding prime factorization not only helps in finding all factors systematically but also plays a crucial role in more complex mathematical domains, such as computing the greatest common divisor (GCD).

Multiples

Contrary to factors, a multiple is formed by multiplying a number by an integer. While factors divide a number, multiples extend the sequence of numbers.

• Generating Multiples: To find multiples of 18, multiply it by integers like 1, 2, 3, and so on; this will yield multiples such as 18, 36, 54...
• While factors of 18 are limited to a finite set (1, 2, 3, 6, 9, 18), its multiples can be infinite.
• Multiples are useful when working with concepts such as least common multiples, which help synchronize events or cycles mathematically.

By understanding both factors and multiples, students can see both the breakdown and building of numbers, gaining a holistic view of number operations.