Determine greatest common factors and least common multiples

For positive whole numbers, the greatest common factor (GCF) is the largest number that divides each quantity exactly, while the least common multiple (LCM) is the smallest positive number that is a multiple of each quantity. These quantities can be determined by listing factors or multiples and by prime factorization: shared prime factors with least exponents produce the GCF, and all required prime factors with greatest exponents produce the LCM. The focus is on positive whole numbers, not negative integers, zero cases, polynomials, or more advanced generalizations.

Detailed Explanation: Determine greatest common factors and least common multiples

The greatest common factor (GCF) is the largest number that divides two numbers exactly. The least common multiple (LCM) is the smallest positive number that both numbers divide into exactly.

Example: Find the GCF and LCM of $(24)$ and $(36)$ .

Step 1: Write each number as a product of prime factors.

24=2^3\cdot 3

36=2^2\cdot 3^2

Step 2: Find the GCF.

For the GCF, use only the prime factors shared by both numbers. Choose the smaller exponent for each shared prime:

\text{GCF}=2^2\cdot 3=4\cdot 3=\boxed{12}

Step 3: Find the LCM.

For the LCM, include every prime factor needed by either number. Choose the greater exponent for each prime:

\text{LCM}=2^3\cdot 3^2=8\cdot 9=\boxed{72}

Check:

$(24\div 12=2)$ and $(36\div 12=3)$ , so $(12)$ is a common factor.
$(72\div 24=3)$ and $(72\div 36=2)$ , so $(72)$ is a common multiple.

Therefore, the GCF is $\boxed{12}$ , and the LCM is $\boxed{72}$ .

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Finding Greatest Common Factor

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