Factors of 28 - Definition, Examples, Quiz, FAQ, Trivia
Learn about factors, prime factorization, and why 28 is a special number
What are Factors of 28?
Factors are numbers that divide evenly into another number without leaving a remainder. For the number 28, the factors are the numbers that can be multiplied together to get 28.
The factors of 28 are: 1, 2, 4, 7, 14, and 28. Let me explain why:
- 1 × 28 = 28
- 2 × 14 = 28
- 4 × 7 = 28
- 7 × 4 = 28
- 14 × 2 = 28
- 28 × 1 = 28
Notice that 28 is a composite number because it has more factors than just 1 and itself. This makes it different from prime numbers which only have two factors.
Key Concept
A factor is a number that divides another number evenly. For 28, the factors are 1, 2, 4, 7, 14, and 28.
How to Calculate Factors Using Division
You can find all the factors of 28 by dividing it by all the integers from 1 to 28 and seeing which divisions give whole numbers without remainders.
Here's how:
| Division | Result | Is Factor? |
|---|---|---|
| 28 ÷ 1 | 28 | Yes |
| 28 ÷ 2 | 14 | Yes |
| 28 ÷ 3 | 9.333 | No |
| 28 ÷ 4 | 7 | Yes |
| 28 ÷ 5 | 5.6 | No |
| 28 ÷ 6 | 4.666 | No |
| 28 ÷ 7 | 4 | Yes |
| 28 ÷ 14 | 2 | Yes |
| 28 ÷ 28 | 1 | Yes |
This method shows us that the only numbers that divide 28 evenly are 1, 2, 4, 7, 14, and 28. These are all the factors of 28.
Remember
Factors always come in pairs! When you find one factor (like 4), you automatically find its pair (7) because 4 × 7 = 28.
Prime Factorization of 28
Prime factorization means breaking down a number into its smallest prime number factors. Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves (like 2, 3, 5, 7, 11).
Let's find the prime factors of 28 using a factor tree:
1. Start with 28
2. Find two factors: 4 and 7 (because 4 × 7 = 28)
3. 7 is already a prime number
4. Break down 4 into 2 × 2 (both prime numbers)
5. So the prime factors are 2 × 2 × 7
We can write this as 2² × 7. This means 28 = 2 × 2 × 7.
Prime Fact
Every composite number can be expressed as a product of prime numbers. This is called the Fundamental Theorem of Arithmetic!
Factor Pairs of 28
Factor pairs are two numbers that multiply together to give the original number. For 28, we have both positive and negative factor pairs.
Positive Factor Pairs:
(1, 28), (2, 14), (4, 7), (7, 4), (14, 2), (28, 1)
Negative Factor Pairs:
(-1, -28), (-2, -14), (-4, -7), (-7, -4), (-14, -2), (-28, -1)
Notice that negative factors come in pairs too because multiplying two negative numbers gives a positive result. So -4 and -7 are factors of 28 because (-4) × (-7) = 28.
The complete list of all factors of 28 is: -28, -14, -7, -4, -2, -1, 1, 2, 4, 7, 14, 28
Pair Tip
When listing factors, start from 1 and work your way up. You'll find pairs automatically!
Why 28 is a Perfect Number
A perfect number is a number that equals the sum of its proper factors. Proper factors are all the factors except the number itself.
Let's check if 28 is perfect:
Proper factors of 28: 1, 2, 4, 7, 14
Sum: 1 + 2 + 4 + 7 + 14 = 28
Since the sum equals 28, it's a perfect number!
28 is actually the second perfect number (the first is 6). Perfect numbers are very rare. The next one is 496!
Perfect numbers have fascinated mathematicians for thousands of years. The ancient Greeks knew about perfect numbers and thought they had special properties.
Math Fact
All even perfect numbers follow a specific pattern discovered by Euclid over 2,000 years ago!
Greatest Common Factor with 28
The Greatest Common Factor (GCF) is the largest factor that two or more numbers share. Let's find the GCF of 28 and another number.
Example: Find the GCF of 28 and 42
Step 1: List factors of both numbers
Factors of 28: 1, 2, 4, 7, 14, 28
Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
Step 2: Find common factors: 1, 2, 7, 14
Step 3: The largest common factor is 14
So GCF(28, 42) = 14
We can also use prime factorization to find GCF:
28 = 2 × 2 × 7
42 = 2 × 3 × 7
The common prime factors are 2 and 7, so GCF = 2 × 7 = 14
GCF Tip
The GCF is always less than or equal to the smaller number. For 28 and 42, the GCF (14) is less than both numbers.
Factors of 28 Practice Quiz
Test your knowledge about factors of 28 with this 5-question quiz. Choose the correct answer for each question.
Frequently Asked Questions
Here are answers to common questions about factors of 28:
Number Trivia
Discover interesting facts about the number 28 and factors:
Ancient Knowledge
The ancient Greeks knew about perfect numbers over 2,000 years ago. They discovered that 6 and 28 were perfect numbers and believed they had special properties.
Lunar Connection
The number 28 is significant in astronomy because the moon takes about 28 days to orbit the Earth. Many ancient calendars were based on 28-day lunar cycles.
Rare Perfection
Perfect numbers like 28 are extremely rare. There are only 51 known perfect numbers, and all currently known perfect numbers are even. No one knows if any odd perfect numbers exist!
February Days
February is the only month that can have 28 days (or 29 in leap years). This makes 28 a special number in our calendar system.
