East Campus, CB 117 361-698-1579 Math Learning Center Finding the GCF and the LCM West Campus, HS1 203 361-698-1860 GREATEST COMMON FACTOR Definition: Greatest Common Factor (GCF) β the largest number/expression that divides into two or more numbers or expressions evenly (no remainder). For example: for the numbers ππππ and ππππ ππ is a common factor but ππ is the greatest common factor, since ππ is the largest number that divides into ππππ and ππππ evenly. Finding the Greatest Common Factor: one approach to finding the GCF is looking at the prime factors that occur the least (look for the smallest exponent) in each of the numbers or expressions that are involved. For instance, in the example using ππππ and ππππ, factor each number into its prime factors. ππππ ππ β’ ππ ππ β’ ππ β’ ππ ππππ β’ ππππ ππππ ππ β’ ππ ππ β’ ππ β’ ππ ππππ The least exponent on the ππ is two and on the ππ it is zero (since 27 doesnβt have any factors of 2) so the GCF is ππππ = ππ Another example of finding the Greatest Common Factor of ππππ and ππππππ: ππππ ππππππ ππ β’ ππ β’ ππ β’ ππ ππ β’ ππ β’ ππ β’ ππ β’ ππ ππππ β’ ππππ β’ ππππ ππππ β’ ππππ β’ ππππ The least exponent of each factor is one so the GCF is ππππ β’ ππππ β’ ππππ = ππππ Examples in Finding the Greatest Common Factor of Algebraic Expressions: The same approach is used to find the GCF of algebraic expressions β factor into prime factors first. Example: find the GCF of ππππππππ ππππ ππ ππππ β ππππ β ππππ β ππππ β ππππ and Choose the least exponent of each factor ππππππππππ ππππ β ππππ β ππππ β ππππ The GCF is ππππ β’ ππππ β’ ππππ = ππππππππ (3, 5, and w are not included because they didnβt occur in both expressions) Example: find the GCF of ππππππ + ππππππ ππππππ (ππ + ππ) ππππππ (ππ + ππ) ππππ β’ ππππ β’ (ππ + ππ)ππ The GCF is ππππ (ππ + ππ)ππ = ππ(ππ + ππ) and ππππππ β ππππ ππ(ππππ β ππ) ππ β’ ππ(ππ β ππ)(ππ + ππ) ππππ β’ ππππ β’ (ππ β ππ)ππ β’ (ππ + ππ)ππ East Campus, CB 117 361-698-1579 Math Learning Center West Campus, HS1 203 361-698-1860 LEAST COMMON MULTIPLE Definition: Least Common Multiple (LCM) β the smallest number/expression that two or more numbers or expressions divide into evenly. For example: for the numbers 6 and 9, 54 is a common multiple but 18 is the least common multiple, since 18 is the smallest number 6 and 9 divides into evenly. Finding the Least Common Multiple: one approach to finding the LCM is looking at the prime factors that occur the most (look for the largest exponent) in each of the numbers or expressions that are involved. For instance, in the example using 6 and 9, factor each number into its prime factors. ππ ππ ππ β’ ππ ππ β’ ππ ππππ β’ ππππ ππππ The largest exponent on the 3 is two and on the 2 it is one so the LCD is 32 β’ 21 = 18 Another example of finding the Least Common Multiple of 90 and 120: ππππ ππππππ ππ β’ ππ β’ ππ β’ ππ ππ β’ ππ β’ ππ β’ ππ β’ ππ ππππ β’ ππππ β’ ππππ ππππ β’ ππππ β’ ππππ Pick the largest exponent on each factor to get the LCM which is ππππ β’ ππππ β’ ππππ = ππππππ Examples in Finding the Least Common Multiples of Algebraic Expressions: The same approach is used to find the LCD (Least Common Denominator) of algebraic expressions β factor the denominator in to prime factors first. Example: find the LCD of ππππππππ ππππ ππ ππππ βππππ βππππ βππππ βππππ and ππππππππππ ππππ βππππ βππππ βππππ Use the largest exponent of each denominator factor, so the LCD is ππππ β’ ππππ β’ ππππ β’ ππππ β’ ππππ β’ ππππ = ππππππππππππππ Example: find the LCM ππππππ + ππππππ ππππππ (ππ + ππ) ππππππ (ππ + ππ) ππππ β’ ππππ β’ (ππ + ππ)ππ and ππππππ β ππππ ππ(ππππ β ππ) ππ β’ ππ(ππ β ππ)(ππ + ππ) ππππ β’ ππππ β’ (ππ β ππ)ππ β’ (ππ + ππ)ππ The LCM is ππππ β’ ππππ β’ ππππ β’ (ππ β ππ)ππ β’ (ππ + ππ)ππ = ππππππ (ππ β ππ)(ππ + ππ)
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