Greatest Common Divisor (GCD) or Greatest Common Factor (GCF)
A. Intersection of Sets:
1.
1) Determine set of all divisors for each given value
2) Determine intersection of all sets obtained
3) GCD is largest value in intersection
and
{
{
}
{
(
2.
,
}
}
)
, and
{
}
}
{
{
}
{ }
(
3.
)
Called “Relatively Prime”
and
{
{
}
}
{
(
}
)
Prof. Fowler
B. Prime Factorization: 1) Determine prime factorization of each given value
2) GCD is product of smallest power of each factor common to all prime
factorizations
1.
and
(
2.
,
)
, and
(
3.
)
and
(
)
Called “Relatively Prime”
C. Euclidean Algorithm:
1.
Divide larger value by smaller value
Divide divisor from Step #1 by remainder from Step #1
Repeat division of divisor by remainder until obtaining remainder of
GCD is divisor that yielded remainder of
and
(
2.
1)
2)
3)
4)
)
and
(
)
Prof. Fowler
Least Common Multiple (LCM)
A. Intersection of Sets:
1.
1) Determine set of all multiples for each given value
2) Determine intersection of all sets obtained
3) LCM is smallest value in intersection
and
{
{
}
{
(
2.
,
}
}
)
, and
{
{
{
}
}
{
(
}
}
)
B. Prime Factorization: 1) Determine prime factorization of each given value
2) LCM is product of largest power of all factors in any factorization
1.
and
(
2.
,
)
, and
(
)
Prof. Fowler
C. Division by [Prime] Factors:
1.
,
1) Divide all values by any common [prime] factor
2) If there is no factor common to all values, divide at least
values by any common [prime] factor, and just carry down all
values that do not share that factor
3) Repeat Steps #1 and #2 until no values have any common
[prime] factors
4) LCM is product of all [prime] factors used and all remaining
[relatively prime] values
, and
Divide by
Divide by
Divide by
:
:
:
,
,
,
,
,
,
(Cannot divide
by
, so just carry
down)
Multiply all divisors and remaining [relatively prime] values:
(
)
OR
Divide by
Divide by
:
:
,
,
,
,
(Cannot divide
by
, so just carry
down)
Multiply all divisors and remaining [relatively prime] values:
(
2.
,
)
, and
Divide by
Divide by
Divide by
Divide by
:
:
:
:
,
,
,
,
,
,
,
,
(Cannot divide
(Cannot divide
(Cannot divide
(Cannot divide
by , so just carry
down)
by , so just carry
down)
by , so just carry
down)
by , so just carry
down)
Multiply all divisors and remaining [relatively prime] values:
(
)
OR
Divide by
Divide by
:
,
,
:
,
,
(Cannot divide
(Cannot divide
by
, so just carry
down)
by , so just carry
down)
Multiply all divisors and remaining [relatively prime] values:
(
)
Prof. Fowler