GCF and LCM Knowledge-Based Study

Study notes

1. What Is a Factor?

   A factor of a whole number is an integer that divides evenly into that number without leaving a remainder. Every positive whole number greater than 1 has at least two factors: 1 and the number itself.

   For example, to list the factors of 24, we find all pairs of whole numbers whose product is 24:

   • 1 and 24
   • 2 and 12
   • 3 and 8
   • 4 and 6

   Therefore, the factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

   Greatest Common Factor (GCF)

   When comparing two or more numbers, a common factor is a number that is a factor of each of them. The Greatest Common Factor (GCF), sometimes called the greatest common divisor, is the largest of these common factors.

   Consider finding the GCF of 18 and 30 by listing factors:

   • Factors of 18: 1, 2, 3, 6, 9, 18
   • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

   The common factors are 1, 2, 3, and 6. The largest among them is 6. Thus, $\text{GCF}(18, 30) = 6$.

   What Is a Multiple?

   A multiple of a whole number is the product of that number and any non-zero whole number. While factors are finite, multiples continue indefinitely.

   For example, the first six non-zero multiples of 6 are:

   6, 12, 18, 24, 30, 36, ...

   Least Common Multiple (LCM)

   A common multiple is a whole number that is a multiple of two or more given numbers. The Least Common Multiple (LCM) is the smallest non-zero whole number that is a multiple of each given number.

   Consider finding the LCM of 8 and 12 by listing multiples:

   • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, ...
   • Multiples of 12: 12, 24, 36, 48, 60, ...

   The common multiples include 24, 48, 72, and so on. The smallest non-zero value is 24. Thus, $\text{LCM}(8, 12) = 24$.

   The Prime Factorization Method

   Listing factors or multiples works well for small numbers, but prime factorization provides a systematic method for larger numbers.

   Every composite number can be uniquely expressed as a product of prime numbers.

   Example: Find the GCF and LCM of 36 and 90

   1. Write the prime factorization of each number in exponential form:
      • $36 = 2^2 \times 3^2$
      • $90 = 2^1 \times 3^2 \times 5^1$
   2. Finding GCF: Take the lowest power of each common prime factor.
      • Common prime factor 2: the lower exponent is $2^1$.
      • Common prime factor 3: the lower exponent is $3^2$.
      • Prime factor 5 is not common to both numbers.
      • $\text{GCF}(36, 90) = 2^1 \times 3^2 = 2 \times 9 = 18$.
   3. Finding LCM: Take the highest power of every prime factor present in either factorization.
      • For 2: the higher exponent is $2^2$.
      • For 3: the higher exponent is $3^2$.
      • For 5: the highest exponent is $5^1$.
      • $\text{LCM}(36, 90) = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180$.

   The Product Rule for Two Numbers

   For any two positive integers $a$ and $b$, the product of their GCF and LCM is equal to the product of the numbers themselves:

   $$\text{GCF}(a, b) \times \text{LCM}(a, b) = a \times b$$

   For example, for 12 and 18:

   • $\text{GCF}(12, 18) = 6$
   • $\text{LCM}(12, 18) = 36$
   • Product of GCF and LCM: $6 \times 36 = 216$
   • Product of numbers: $12 \times 18 = 216$

   This identity can be used to quickly find one quantity if the other three are known: $$\text{LCM}(a, b) = \frac{a \times b}{\text{GCF}(a, b)}$$

   When to Use GCF vs. LCM

   • Use GCF when you need to split or divide objects into equal groups, find the largest identical partitions, or simplify fractions to their simplest form.
   • Use LCM when you need to find when repeated events synchronize or coincide, find common denominators to add or subtract fractions, or determine when two repeating patterns align.