# Counting Factors,Greatest Common Factor,Least Common Multiple

# Counting Factors, Greatest Common Factor , Least Common Multiple

•Every number can be **expressed in terms of primes **

i.e. where n2, n3, n5, n11 are integer
exponents (maybe 0)

•*Example. Express 144 in terms of its prime
factorization*

144 = 2^{4} •3^{2} . So n2 = 4, n3 = 2, and all the others are 0

•The **number of factors** for a given whole number is related to the
exponents in its prime

factorization. The number of factors = (n2 +1)(n 3+1)(n 5+1)…

*•Example. How many factors are there for 144?
*Since the factorization for 144 is 2

^{4}•3

^{2}, there are (4+1)(2+1) = 15 different factors for 144

•You find these factors by taking all possible combinations of the prime factorization with exponent

values from 0 to n2, 0 to n3, 0 to n5, etc

*•Example. What are the factors of 144?
*

*•Example, page 208 number 1d. How many factors does 12 ^{4}
have?
*12

^{4}= (2

^{2}•3)

^{4}= 2

^{8}•3

^{4}. It has (8+1)(4+1) = 45 factors

*•Example, page 208 number 2a. Factor 120 into primes
*120 = 60 •2 = 30•2

^{2}= 10 •2

^{2}= 5•3•2

^{3}

__ The Greatest Common Factor:__•The

**greatest common factor (GCF)**of two (or more) nonzero whole numbers is the largest whole

number that is a factor of both (all) of the numbers

•You can find the greatest common factor by the

**set intersection method**

-Finding all factors of each of the numbers and placing them in sets

-Finding the intersection of those sets

-Finding the largest value in the intersection

•Example, page 208 number 6d. Find the GCF(42, 385)

Factors of 42:

Factors of 385:

GCF(42, 385) = 7

•You can also find the GCF by the prime factorization
method

-Find the prime factorization of each number

-Take whatever they have in common (to the highest power possible )

*•Example, page 208 number 6d. Find the GCF(42, 385)
*Factorization 42 = 2•1= 2•3•7

Factorization 385 = 5•7 = 5• 11

GCF(42, 385) = 7

^{1}= 7

*•Example, page 208 number 6f. Find the GCF(338, 507)
*Factorization 338 = 2•69 = 2 •3

^{2}

Factorization 507 = 3•69 = 3•3

^{2}

GCF(338, 507) = 13

^{2}= 169

•If a and b are whole numbers with a ≥ b , then GCF(a, b) = GCF(a - b, b)

*•Example, page 208 number 6f (again)
*GCF(507, 338) = GCF(507 - 338, 338) = GCF(169, 338)

__ Least Common Multiple:__•The

**least common multiple (LCM)**of two (or more) nonzero whole numbers is the smallest

nonzero whole number that is a multiple of each (all) of the numbers

•You can find the least common multiple with the **set
intersection method **

-List the nonzero multiples of each number

-Intersect the sets

-Take the smallest element in the intersection

*•Example, page 208 number 7d. Find the LCM(66, 88)
*Multiples of 88 {88, 176, 264, 352…}

Multiples of 66 {66, 132, 198, 264, 330…}

LCM(66, 88) = 264

•You can also find the LCM with the prime factorization
method (similar to the buildup method)

-Express the numbers in their prime factorization

-Take each factor (to its highest power ) from each factorization, but do not
repeat

*•Example, page 208 number 7d (again)
*88 = 2 •44 = 2

^{2}•2

^{2}= 2

^{3}•11

66 = 3•2 = 2•11

LCM(66, 88) = 2

^{3}•11 = 264

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