Problem 88 - Fundamentals of Mathematics

Question

Find all factors of 24 .

Step-by-Step Solution

Answer

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

Step 1: Understand the concept of factors

Factors are numbers that divide the given number without leaving any remainder. In other words, if you can divide the number by something and the result is a whole number, it's a factor.

Step 2: Start with the smallest factors

Any whole number n is divisible by 1 and itself, so we start by listing the smallest factor, which is 1. Hence, 1 is a factor of 24. Similarly, the number itself, 24, is a factor.

Step 3: Test divisibility by small whole numbers

To find other factors, begin testing if small whole numbers divide into 24 evenly. After 1, test 2 and observe (24 ÷ 2 = 12), hence, 2 and 12 are factors. Continue with 3 (24 ÷ 3 = 8), so 3 and 8 are factors. With 4 (24 ÷ 4 = 6), so 4 and 6 are factors.

Step 4: List all factors and stop when smaller pairs are tested

All factors are found when small numbers are paired with larger numbers that result from division. For number 24, after listing and testing 4, continuing results in repeating numbers. These factors will be 1, 2, 3, 4, 6, 8, 12, and 24.

Key Concepts

Divisibility Rules

Understanding the concept of divisibility rules is fundamental to determining the factors of a number. Divisibility rules are shortcuts that help us quickly determine if one number can divide another without a remainder. Instead of manually dividing numbers, these rules simplify our work. For example:

• A number is divisible by 2 if it is even. That means it ends in 0, 2, 4, 6, or 8.
• If the sum of a number's digits is divisible by 3, then the number itself is divisible by 3 as well.
• A number is divisible by 5 if it ends in 0 or 5.

These rules make it faster to assess potential factors. When finding factors of 24, we used the rule for 2, as 24 is even, making 2 a factor. Similarly, the sum of digits of 24 is 6 (2 + 4 = 6), which is divisible by 3, identifying 3 as a factor. Applying these rules can greatly expedite the factorization process.

Least Common Multiple

When discussing factors, the concept of the Least Common Multiple (LCM) often accompanies it. The LCM of two numbers is the smallest number, greater than zero, that is a multiple of both numbers. Understanding the LCM is valuable for solving problems involving fractions or when finding common denominators.
Consider finding the LCM of 6 and 8. List the multiples of each:

• Multiples of 6: 6, 12, 18, 24, 30, 36...
• Multiples of 8: 8, 16, 24, 32, 40...

The smallest multiple common to both lists is 24, making it the LCM.
Finding the LCM involves looking for the overlap in multiples. It often involves using prime factors, multiplying each prime factor the greatest number of times it appears in any of the numbers. This method ensures that you only get the necessary shared multiples without missing any.

Prime Factorization

Prime factorization breaks down a number into its most basic building blocks: prime numbers. A prime number has only two factors, itself and one.
The process of prime factorization involves dividing the original number by the smallest prime number and continuing the division with the quotient until the quotient itself is a prime.
For the number 24, we start with the smallest prime number, 2:

• 24 ÷ 2 = 12
• 12 ÷ 2 = 6
• 6 ÷ 2 = 3
• Finally, 3 is a prime number.

After this division process, we find that 24 = 2 × 2 × 2 × 3 or in exponential form, 24 = 2^3 × 3.
Prime factorization is crucial for various applications, such as simplifying fractions and finding the greatest common divisor (GCD) or least common multiple (LCM) of numbers. With practice, this method becomes a straightforward process for simplifying complex numerical problems.