The GCD (greatest common divisor) often needs to be calculated for two or more numbers. There are several established methods used to calculate the GCD, but the method in this lesson plan can be expanded to find the GCD of three or more numbers, and is easier to understand for basic level math studies.
What Is The Greatest Common Factor?
The greatest common factor of two numbers is the largest positive whole number that can be divided into each of the two numbers without leaving a remainder. It is also known as the highest common factor or highest common divisor. Any two numbers, A and B, are related to their LCM and GCD by the formulas:
- GCD = A × B ÷ LCM
- LCM = A × B ÷ GCD
So it is only necessary to calculate one out of the LCM or the GCD of any pair of numbers, since the formulas may be used to calculate the other.
Prime Factors and Factorization
The first step to be taken when finding the GCD of a set of numbers is to factorize each of the numbers into its prime factors. This process is described in the article "Finding The Prime Factors Of A Number". Not only is this the first step in the process, but it is actually most of the process too. When the prime factors have been found, calculating the highest common factor is a quick and easy process.
Calculating Greatest Common Divisor
Figure 1 shows the prime factors of 14 and 24 visually. The highest common factor of 24 and 14 may be described as the largest number that divides each number without leaving a remainder. Looking at the primes factors of 14 and 24 in Figure 1, it an be seen that:
14 = 2 × 7 and
24 = 2 × 2 ×2 × 3
Taking each prime factor in turn for each number
2: This goes into 14 once, and goes into 24 three times. So the highest common factor has one factor of 2 in it.
3: This does not divide into 14 at all, and divides into 24 once. So the highest common factor has no factor of 3 in it.
5: This does not divide into 14 or 24, so the highest common factor has no 5 in it.
7: This divides 14 once, and does not divide 24 at all. 7 is therefore not a factor of the highest common factor.
The highest common factor, or greatest common divisor (GCD) of 14 and 24 is therefore 2.
Least Common Multiple Lesson Plan Summary
Least common multiple is useful when teaching and manipulating fractions. The process to find the LCM for two or more numbers at a time involves knowing the first few prime numbers, then finding the prime factors for each number, deducing the prime factors of the LCM, and multiplying these together to find the LCM itself. For any pair of numbers it is not necessary to calculate the LCM as well as the GCD, as they are related by a simple formula.
Least Common Multiple and Prime Factorization References
Most of the information here is widely known and available, although the CD " Math Skill Builders: Set 3: Finding Multiples " is a useful resource.
Martin Bell - Martin holds a B.Sc. degree in chemical engineering, and an M.Sc. degree in electronics and computing. He has spent more than 25 years ...
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