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Whenever we multiply two whole numbers, we get another number. The numbers we multiplied to get the product are called the factors of the product. For example, \(2 \times 3 = 6\). This implies that 2 and 3 are the factors of 6. Another conclusion we can draw from this is that factors of a number completely divide the number without leaving any remainder.

Example: Let’s consider the number 24. Now, 24 can be divided into factors 6 and 4. Also, 6 can be further factorized into 3 and 2. Moreover, 4 can also be factorized into 2 and 2. So, from this we can see that the other factors of 24 are 3, 8, and 2. This is because \(12 \times 2 = 8 \times 3 = 3 \times 8 = 24\).

Some Facts about Factors:

  • Each and every number has a smallest factor which is 1.
  • Every number has a minimum of two factors, which are 1 and the number itself.
  • Numbers which have only two factors (1 and the number itself) are called prime numbers.

Prime Factorization

Prime factorization is defined as the product of all the prime factors of a number whose multiplication gives the number itself. To write the prime factors of a number, we may have to repeat the number. For example, the factors of 8 are 1 and 2, but to represent 8, we use:

  • \(8 = 2 \times 2 \times 2\)

prime_factors_of_8

Greatest Common Factor (GCF)

El GCF es el factor más grande que dos o más números tienen en común. Para encontrar el GCF de dos números, sigue estos pasos:

  1. Descompone cada número en sus factores primos.
  2. Identifica los factores primos comunes.
  3. Multiplica los factores comunes para obtener el GCF.

Ejemplo: Encuentra el GCF de 100 y 50.

  • 100 = 2 × 2 × 5 × 5
  • 50 = 2 × 5 × 5

Por lo tanto, los factores comunes son 2 y 5 × 5. Entonces, el GCF de 100 y 50 es:

  • GCF(100, 50) = 2 × 5 × 5 = 50

Exercises for Finding the Greatest Common Factor (GCF)

  1. GCF(33, 45) =
  2. GCF(26, 14) =
  3. GCF(11, 14) =
  4. GCF(12, 42) =
  5. GCF(45, 42) =
  6. GCF(42, 34) =
  7. GCF(26, 52) =
  8. GCF(90, 27) =
  9. GCF(92, 80) =
  10. GCF(27, 18) =

Answers

  1. GCF(33, 45) = 3

    Solution:

    Step 1: List the prime factors of each number: \(33 = (3, 11)\) and \(45 = (3, 3, 5)\)

    Step 2: Identify the common prime factors. The common factor in these two numbers is 3. Therefore, their greatest common factor is 3.

  2. GCF(26, 14) = 2

    Solution:

    Step 1: List the prime factors of each number: \(26 = (2, 13)\) and \(14 = (2, 7)\)

    Step 2: Identify the common prime factors. The common factor in these two numbers is 2. Therefore, their greatest common factor is 2.

  3. GCF(11, 14) = 1

    Solution:

    Step 1: List the prime factors of each number: \(11 = (11)\) and \(14 = (2, 7)\)

    Step 2: Identify the common prime factors. There are no common prime factors, so the greatest common factor is 1.

  4. GCF(12, 42) = 6

    Solution:

    Step 1: List the prime factors of each number: \(12 = (2, 2, 3)\) and \(42 = (2, 3, 7)\)

    Step 2: Identify the common prime factors. The common factors are 2 and 3. Therefore, their greatest common factor is \(2 \times 3 = 6\).

  5. GCF(45, 42) = 3

    Solution:

    Step 1: List the prime factors of each number: \(45 = (3, 3, 5)\) and \(42 = (2, 3, 7)\)

    Step 2: Identify the common prime factors. The common factor is 3. Therefore, their greatest common factor is 3.

  1. GCF(42, 34) = 2

    Solution:

    Step 1: List the prime factors of each number: \(42 = (2, 3, 7)\) y \(34 = (2, 17)\)

    Step 2: Identify the common prime factors. The common factor is 2. Therefore, their greatest common factor is 2.

  2. GCF(26, 52) = 26

    Solution:

    Step 1: List the prime factors of each number: \(26 = (2, 13)\) y \(52 = (2, 2, 13)\)

    Step 2: Identify the common prime factors. The common factors are 2 and 13. Therefore, their greatest common factor is \(2 \times 13 = 26\).

  3. GCF(90, 27) = 9

    Solution:

    Step 1: List the prime factors of each number: \(90 = (2, 3, 3, 5)\) y \(27 = (3, 3, 3)\)

    Step 2: Identify the common prime factors. The common factors are 3 and 3. Therefore, their greatest common factor is \(3 \times 3 = 9\).

  4. GCF(92, 80) = 4

    Solution:

    Step 1: List the prime factors of each number: \(92 = (2, 2, 23)\) y \(80 = (2, 2, 2, 2, 5)\)

    Step 2: Identify the common prime factors. The common factors are 2 and 2. Therefore, their greatest common factor is \(2 \times 2 = 4\).

  5. GCF(27, 18) = 9

    Solution:

    Step 1: List the prime factors of each number: \(27 = (3, 3, 3)\) y \(18 = (2, 3, 3)\)

    Step 2: Identify the common prime factors. The common factors are 3 and 3. Therefore, their greatest common factor is \(3 \times 3 = 9\).