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Finding the Greatest Common Factor (GCF) Using Short Division

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Finding the Greatest Common Factor (GCF)

The greatest common factor (GCF) of two or more numbers is the largest number that divides evenly into all of them. Finding the GCF can be helpful in simplifying fractions, solving problems involving ratios, and understanding the relationships between numbers.

One way to find the GCF is by using the short division method. This method involves dividing the numbers by their common factors until no further division is possible. The GCF is then the product of the common factors.

Steps to Find the GCF Using Short Division

  1. Write the numbers you want to find the GCF of in a row.
  2. Find a common factor of all the numbers. Divide each number by that factor.
  3. Write the quotients below the original numbers.
  4. Repeat steps 2 and 3 until no further division is possible.
  5. The GCF is the product of all the common factors used in the division.

Example

Let's find the GCF of 12, 18, and 24.

  1. Write the numbers in a row: 12, 18, 24
  2. Find a common factor. In this case, 2 is a common factor. Divide each number by 2: 12 ÷ 2 = 6, 18 ÷ 2 = 9, 24 ÷ 2 = 12.
  3. Write the quotients below the original numbers:</n 
      2 | 12  18  24
      6   9  12
  4. Repeat steps 2 and 3. We can divide again by 2: 6 ÷ 2 = 3, 12 ÷ 2 = 6. We can't divide 9 by 2, so we leave it as is.
  5. Write the quotients below the previous row:
    6.   2 | 12  18  24
      6   9  12
  2 | 3   9   6
      3   9   3
  6. Repeat steps 2 and 3. We can divide again by 3: 3 ÷ 3 = 1, 9 ÷ 3 = 3, 3 ÷ 3 = 1.
  7. Write the quotients below the previous row:
    8.   2 | 12  18  24
      6   9  12
  2 | 3   9   6
      3   9   3
  3 | 1   3   1
      1   1   1
  8. We can't divide any further. The common factors used in the division are 2, 2, and 3.
  9. The GCF is the product of these common factors: 2 × 2 × 3 = 12.

Therefore, the GCF of 12, 18, and 24 is 12.

Conclusion

The short division method is a simple and effective way to find the GCF of two or more numbers. It involves dividing the numbers by their common factors until no further division is possible. The GCF is then the product of the common factors.

GCF