Lcm

2. The Least Common
Multiple (L.C.M.) of
two positive integers is
the
smallest positive integer
that is a multiple of both.

3. The Least Common
Multiple of two
integers a and b is
denoted by
𝑎, 𝑏

4. Ways of getting the LCM
1. LISTING METHOD
2. FACTOR TREE METHOD
3. CONTINUOUS
DIVISION METHOD

5. 1. Listing Method
First Step: List the multiples
Second Step: List the common multiples
Common Multiples = 24, 48, 72
Least Common Multiples (LCM) = 24
8
12

6. 2. Continuous Division Method
2
2
Least Common Multiple = 24

7. 3. Factor Tree Method
8 = 𝟐𝟑
8 12
2
4
2 2
3
4
2 2
12 = 𝟐𝟐
𝒙 𝟑
𝟐𝟑
𝒙 𝟑 = 𝟐𝟒 LCM = 24

8. Example : 𝟒𝟎, 𝟖𝟎
40 80
2
20
10 2
2
40
20 2
𝟐𝟒
𝒙 𝟓
𝟐𝟑
𝒙 𝟓 16 x 5 = 80
5 2 10 2
LCM

9. We can figure out 𝑎, 𝑏 once we
have the prime factorization of a
and b. To do that, let
𝑎 = 𝑝1
𝑎1
𝑝2
𝑎2
… . 𝑝𝑚
𝑎𝑛
and 𝑏 = 𝑝1
𝑏1
𝑝2
𝑏2
… . 𝑝𝑚
𝑏𝑛
,
Where (as above) we exclude any prime with 0
power in both 𝑎 𝑎𝑛𝑑 𝑏. 𝑇ℎ𝑒𝑛 𝑎, 𝑏 =
𝑝1
max(𝑎1𝑏1)
𝑝1
max(𝑎2𝑏2)
… 𝑝𝑚
max 𝑎𝑛𝑏𝑛
, 𝑤ℎ𝑒𝑟𝑒 max(𝑎, 𝑏) is
the maximum of two integers a and b.

10. THEOREM THAT
RELATES THE
LCM OF TWO
POSITIVE
INTEGERS TO
THEIR GCD

11. Theorem
Let a and b be two positive integers. Then
1. 𝑎, 𝑏 ≥ 0 ;
2. 𝑎, 𝑏 = ab/(a, b);
3. If 𝑎 𝑚 𝑎𝑛𝑑 𝑏 𝑚, then 𝑎, 𝑏 𝑚;

12. Proof
The proof of part 1
𝑎, 𝑏 ≥ 0 ;
follows from the
definition.

13. Proof
As for part 2,
𝑎 = 𝑝1
𝑎1
𝑝2
𝑎2
… . 𝑝𝑚
𝑎𝑛
and 𝑏 = 𝑝1
𝑏1
𝑝2
𝑏2
… . 𝑝𝑚
𝑏𝑛
,
Notice that since,
𝒂, 𝒃 = 𝒑𝟏
𝐦𝒊𝒏(𝒂𝟏𝒃𝟐)
𝒑𝟐
𝐦𝒊𝒏(𝒂𝟐𝒃𝟐)
… 𝒑𝒏
𝐦𝒊𝒏(𝒂𝒏𝒃𝒏)
and
𝑎, 𝑏 = 𝑝1
m𝑎𝑥(𝑎1𝑏1)
𝑝2
m𝑎𝑥(𝑎2𝑏2)
… 𝑝𝑛
m𝑎𝑥(𝑎𝑛𝑏𝑛)
then
𝑎, 𝑏 = ab/(a, b);
Let

14. 𝑎, 𝑏 𝑎, 𝑏 = 𝑝1
m𝑎𝑥(𝑎1𝑏1)
𝑝2
m𝑎𝑥(𝑎2𝑏2)
… 𝑝𝑛
m𝑎𝑥(𝑎𝑛𝑏𝑛)
𝑝1
m𝑖𝑛(𝑎1𝑏2)
𝑝2
m𝑖𝑛(𝑎2𝑏2)
… 𝑝𝑛
m𝑖𝑛(𝑎𝑛𝑏𝑛)
= 𝑝1
𝑚𝑎𝑥 𝑎1𝑏1 +m𝑖𝑛(𝑎1𝑏1)
𝑝2
𝑚𝑎𝑥 𝑎2𝑏2 +m𝑖𝑛(𝑎2𝑏2)
… 𝑝𝑚
𝑚𝑎𝑥 𝑎𝑛𝑏𝑛 +m𝑖𝑛(𝑎𝑛𝑏𝑛)
= 𝑝1
𝑎1+𝑏1
𝑝2
𝑎2+𝑏2
… 𝑝𝑚
(𝑎𝑛+𝑏𝑛)
= 𝑝1
𝑎1
𝑝2
𝑎2
… 𝑝𝑚
𝑎𝑛
𝑝1
𝑏1
𝑝2
𝑏2
… 𝑝𝑚
𝑏𝑛
= ab

15. LCM = Product of the numbers / GCD
Numbers Product GCD LCM
Example:
2 , 8
9 , 24
6 , 12
16
216
72
2
3
6
16 ÷ 2 = 8
216 ÷ 3 = 72
72 ÷ 6 = 12

16. For part 3
If 𝑎 𝑚 𝑎𝑛𝑑 𝑏 𝑚, then 𝑎, 𝑏 𝑚;
Example 1:
2, 8 ; 2, 8 = 8
2 16 , 8 16, 𝑡ℎ𝑒𝑛 8 16
Solution:
Example 2:
6, 9 ; 6, 9 = 18
Solution:
6 6 , 9 6, 𝑡ℎ𝑒𝑛 18 6