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ALGEBRA
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GCD INTRO
THE DEFINITION of ’LARGER’
In this section, we will be talking about larger numbers and smaller numbers, thus it seems appropriate
that we take a moment here to provide a sound definition of larger. We will declare an integer number
a larger than integer, b, and we write a > b, if and only if the quantity a − b is a positive integer (we
sometimes write positive integers as Z+ ). Moreover, to say a is larger than b is the same as saying b is
less than a and is the same as saying a is greater than b, and it’s the same as saying b < a. Symbolically,
we write for integers a and b, the definition is :
a > b ⇐⇒ a − b ∈ Z+
Also note this definition can be extended beyond the integers and to general real numbers by defining
a < b if b − a ∈ R+ .
a working UNDERSTANDING of GREATEST COMMON DIVISOR
Divisor is another word for factor. It does not always happen this way but in this case it does: The
definition of the phrase is very much consistent with the expected definition of the phrase. The greatest
common divisor of two (or more) numbers is precisely the largest of the common divisors of the two
numbers. We often use gcd(a, b) to represent the greatest common divisor of the numbers a and b. One
way to find the gcd for two numbers is to actually list every divisor of each of the numbers, identify all
the common ones, and select the greatest of the common divisors. For example, to find the gcd(24, 36)
we first list all the divisors of 24 and all the divisors of 36:
divisors of 24 : 1, 2, 3, 4, 6, 8, 12, 24
then
divisors of 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36
Then we simply select the largest of the common divisors, 12. While most of the time this definition
serves us well, there may be times when it is simply not adequate. In algebra, we will frequently work
with variables, and determining whether x2 y 3 or y 2 x3 is bigger may be dubious, thus we offer a second,
more complete definition of the gcd of two numbers. This definition is based on the prime factorization
of the two number.
THE DEFINITION of GREATEST COMMON DIVISOR
This is a more general definition of the gcd(a, b). This definition depends on the prime factorization of
a and b. The idea is that the prime factorization really exposes the composition of each number. Once
exposed, we can just look at the numbers and see what is common. For example, to find gcd(24, 36) we
first prime factorize them to write
gcd(24, 36) = gcd(23 3, 22 32 )
This exposes 24 and shows what it is really made up of. The number 24 is made of three 2’s and one 3.
Similarly, 36 is made up of two 2’s and two 3’s. Now we try to make sense of what they have in common.
The most that they have in common is two 2’s and one 3. Indeed, their gcd is 22 · 3 = 12 as expected.
We can not state the definition of gcd(a, b) formally.
gcd(a, b) = the product of all common prime divisors, including multiplicity
EXAMPLE of GREATEST COMMON DIVISOR
Find gcd(36, 90)
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2007
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ALGEBRA
Sec. 04
MathHands.com
Márquez
SOLUTION:
We first prime factorize each number:
36 =
=
=
90 =
=
=
6·6
2·3·2·3
22 · 32
9 · 10
3·3·2·5
2 · 32 · 5
TT
BI
BI
TT
BI
BI
Therefore, the common factors are 2 and 3 with 3 occurring twice. Thus
gcd(36, 90) = gcd(22 · 32 , 2 · 32 · 5) = 2 · 32 = 18
EXAMPLE of GREATEST COMMON DIVISOR
Find gcd(36x5 y 3 , 90y 7 x4 z 2 )
SOLUTION:
We first prime factorize each term, to get
36x5 y 3 = 22 · 32 · x5 · y 3
90y 7 x4 z 2 = 2 · 32 · 5 · y 7 · x4 · z 2
BI
BI
Therefore,
gcd(36x5 y 3 , 90y 7 x4 z 2 ) = 2 · 32 x4 y 3 = 18x4 y 3
1. (TRUE or FALSE) Use the definition determine if the statement is true or false.
20 > 3
Solution: TRUE, since 20 − 3 is a positive integer.
2. (TRUE or FALSE) Use the definition determine if the statement is true or false.
3 > −5
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ALGEBRA
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Solution: TRUE, since 3 − (−5) = 8 is a positive integer.
3. (TRUE or FALSE) Use the definition determine if the statement is true or false.
−3 > −5
Solution: TRUE, since −3 − (−5) is a positive integer.
4. (TRUE or FALSE) Use the definition determine if the statement is true or false.
0 > −5
Solution: TRUE, since 0 − (−5) is a positive integer.
5. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(26, 14)
Solution: gcd(26, 14) = 2
6. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(26, 36)
Solution: gcd(26, 36) = 2
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ALGEBRA
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7. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(126, 360)
Solution: gcd(126, 360) = 18
8. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(426, 360)
Solution: gcd(426, 360) = 6
9. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(76, 96)
Solution: gcd(76, 96) = 4
10. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(42, 84)
Solution: gcd(42, 84) = 42
11. List all divisors, identify the common divisors, then find the indicated gcd.
gcd(42, 126)
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ALGEBRA
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Solution: gcd(42, 126) = 42
12. Prime factorize each of the terms, then find the indicated gcd.
gcd(75, 100, 125)
Solution: 25
13. Prime factorize each of the terms, then find the indicated gcd.
gcd(75x5 , 100x3 )
Solution: 25x3
14. Prime factorize each of the terms, then find the indicated gcd.
gcd(75x5 y 5 , 100x3 y 8 )
Solution: 25x3 y 5
15. Prime factorize each of the terms, then find the indicated gcd.
gcd(x5 y 5 t9 , t3 y 8 )
Solution: t3 y 5
16. Prime factorize each of the terms, then find the indicated gcd.
gcd(x5 y 5 t9 , t3 y 8 x2 )
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ALGEBRA
Sec. 04
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Solution: x2 t3 y 5
17. Find the gcd(x, 2x), where x is a natural number.
Solution: x, if prime factorized, x would have the exact same primes as 2x with the exception of
an extra ’2’ in the prime factorization of 2x. Thus, the largest number of common primes would be
exactly all primes in x counting multiplicity.
18. Find the gcd(x, x2 ), where x is a natural number.
Solution: x, if prime factorized, x would have the exact same primes as x2 with the exception x2
would have twice as many of each. Thus, the largest number of common primes would be exactly all
primes in x counting multiplicity.
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