Fundamentals of Mathematics
1.3 Fractions
Ricky Ng
Lecture 4
September 4, 2013
Ricky Ng
Fundamentals of Mathematics
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Ricky Ng
Fundamentals of Mathematics
1.3 Fractions
There are three common operations we encounter when dealing
with fractions:
Greatest common divisor and Least common
multiple
Addition / Subtraction
Multiplication / Division
Ricky Ng
1.3 Fractions
Greatest Common Divisor
Definition (gcd)
Given natural numbers n and m, the greatest common
divisor (GCD) is the largest factor of both n and m. We
denote it by gcd(n, m).
Example: Find gcd(24, 36).
Ricky Ng
1.3 Fractions
A direct method to find GCD (especially with large numbers) is
prime factorization.
Theorem (Prime Factorization)
Every integer can be written as a product of powers of
primes.
Example: Find the prime factorizations of 36 and 60, then find
gcd(36, 60).
Ricky Ng
1.3 Fractions
Find gcd(12, 64).
Ricky Ng
1.3 Fractions
When dealing with large numbers, sometime we leave the gcd in
prime factorization form. Find gcd(72, 128).
Ricky Ng
1.3 Fractions
Least Common Multiple
Definition (lcm)
Given two integers n and m, the least common multiple
(lcm) is the smallest integer that is divisible by both n and m.
We denote this number by lcm(n, m).
Note that a multiple always exists, namely n × m. However,
we’d like to get the smallest one ≤ n × m.
Example: Find lcm(12, 18).
Ricky Ng
1.3 Fractions
A direct method method to find LCM also bases on the prime
factorization.
Rule (For LCM)
1
Find the prime factorizations of m and n.
2
Find gcd(m, n) identify the remaining prime factors of m
and n.
3
Multiply gcd(m, n) and the remaining prime factors.
Example: Find the prime factorizations of 36 and 60, then find
lcm(36, 60).
Ricky Ng
1.3 Fractions
Ricky Ng
1.3 Fractions
Examples
Find lcm(72, 192).
Ricky Ng
1.3 Fractions
Find lcm(26, 65).
Ricky Ng
1.3 Fractions
Examples
Find gcd(48, 80) and lcm(48, 80).
Ricky Ng
1.3 Fractions
What if we want to find LCM of more than two numbers? Find
lcm(4, 6, 9).
Ricky Ng
1.3 Fractions
Find gcd(16, 24, 36) and lcm(16, 24, 36).
Ricky Ng
1.3 Fractions
Addition/Subtraction
The rules for addition and subtraction of integers extend to
fractions, when they share the same denominators.
Rule
a±b
a b
± =
,
c c
c
where c 6= 0.
Example:
4
3
+ 53 .
Ricky Ng
1.3 Fractions
Examples
Find 3 43 + 2 14 .
Find 8 87 − 6 58 .
Ricky Ng
1.3 Fractions
Find 1 52 − 3 45 . (Hint: First change to improper forms.)
Ricky Ng
1.3 Fractions