What are the factors of 28

General
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what are the factors of 28

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What are the factors of 28?

Answer:

To determine the factors of 28, we need to identify all the whole numbers that can divide 28 without leaving a remainder. Factors are the numbers you multiply together to get another number.

Step-by-Step Process to Find Factors

  1. Start with 1 and the Number Itself:

    • The number 1 is a universal factor for any integer, and any number is a factor of itself. So, 1 and 28 are factors of 28.

  2. Divide 28 by Subsequent Whole Numbers:

    • Check the divisibility of 28 by whole numbers starting from 2 upwards.

  3. Pair the Factors:

    • If a whole number divides 28 evenly, pair it with the quotient that results from the division. For instance, if 28 ÷ 2 equals 14, then 2 and 14 are a factor pair.

Detailed Factorization Process

  • 28 ÷ 1 = 28: As mentioned before, 1 and 28 are factors.
  • 28 ÷ 2 = 14: Since 28 divided by 2 gives a whole number (14), both 2 and 14 are factors.
  • 28 ÷ 3: This gives approximately 9.33, which is not a whole number, so 3 is not a factor.
  • 28 ÷ 4 = 7: Since 28 divided by 4 gives a whole number (7), both 4 and 7 are factors.
  • 28 ÷ 5: This gives approximately 5.6, which is not a whole number, so 5 is not a factor.
  • 28 ÷ 6: This gives approximately 4.67, which is not a whole number, so 6 is not a factor.

Final List of Factors

After going through the process, the factors of 28 are:

  • 1, 2, 4, 7, 14, and 28

These numbers are all the factors that can multiply in pairs to result in the product 28. Here are the factor pairs:

  • 1 × 28 = 28
  • 2 × 14 = 28
  • 4 × 7 = 28

Special Characteristics of 28

  • Perfect Number: Interestingly, 28 is what’s called a perfect number. A perfect number is a number that is equal to the sum of its proper divisors (excluding itself). For 28, the divisors excluding 28 itself are 1, 2, 4, 7, and 14. Their sum is (1 + 2 + 4 + 7 + 14 = 28).

  • Even Number: Being divisible by 2, 28 is classified as an even number.

Prime Factorization of 28

Prime factorization involves breaking a number down into its prime number components. For 28, this can be shown as:

  • (28 \div 2 = 14)
  • (14 \div 2 = 7)
  • 7 is a prime number, so division stops here.

Therefore, the prime factorization of 28 is:

  • 2^2 \times 7

Verification

We can recombine these prime factors to ensure that 28 is the correct product:

  • (2 \times 2 \times 7 = 28)

This confirms that the factorization is correct.

Conclusion

The factors of 28 encompass both the trivial factors (1 and 28 itself) and non-trivial ones (2, 4, 7, and 14), while its prime factorization gives insight into its composition solely in terms of prime numbers.

This in-depth exploration not only provides the factors of 28 but also illustrates how it uniquely fits the category of a perfect number. Whether for mathematical studies or practical problem-solving, understanding such factorization can be enormously helpful.