This document discusses methods for finding the lowest common multiple (LCM) of two or more numbers. The old method involves listing all multiples of each number and taking the first number that appears in both lists. The new, better method breaks down each number into prime factors, takes the highest exponent of any repeated factors, and multiplies all factors together. For example, to find the LCM of 30 and 8 using the new method: 30 breaks down to 2 × 3 × 5 and 8 to 2 × 2 × 2. The highest exponent of 2 is 3, so the LCM is 3 × 5 × 23 = 120.

1. Lowest Common Multiple of a
number
Richie O’Connor

2. What is the ‘Lowest Common
Multiple’?
• It is the lowest of the common multiples
of any numbers.
• So, we make lists of the multiples of
both numbers and take the first number
which appears in both lists.

3. There are 2 methods2 methods to find the
Lowest Common Multiple (L.C.M):
1.The old method.
2.The new method which is better.

4. 1. The old method
a) Write the list of multiples of every number
b) Take the common multiples
c) Choose the lowest

5. For example:
• Find the Lowest Common Multiple (L.C.M.) of 1010
and 15.15.
• A)A) List the multiples of both numbers:List the multiples of both numbers:
– Multiples of 10Multiples of 10 →→ 10, 20,10, 20, 3030, 40, 50,, 40, 50, 6060, 70, 80,, 70, 80, 9090……
– Multiples of 15Multiples of 15 →→ 15,15, 3030, 45,, 45, 6060, 75,, 75, 9090……

6. • B) Take the common multiples:B) Take the common multiples:
– Common multiples →→ 3030 –– 6060 –– 90…90…
• C) Choose the lowest:C) Choose the lowest:
– TheThe llowestowest ccommonommon mmultiple of 10 and 15 isultiple of 10 and 15 is 3030
– →→ L.C.M. (10, 15) = 30L.C.M. (10, 15) = 30

7. But…But…
• This method does not work well with big
numbers.
• The 2nd
method is better.

8. 2) The new method
a) Break down both numbers into their prime
factors.
b) Take all the factors.
c) If a factor is repeated, take the one with the
highest exponent.

9. For example:
• Find the Lowest Common Multiple (L.C.M.) of
3030 and 8.8.
A. Break down both numbers into their prime factors.
30 │30 │ 22 8 │8 │ 22
15 │15 │ 33 4 │4 │ 22
5 │5 │ 55 2 │2 │ 22
1 │1 │ 1 │1 │

10. B) Take all the factors.
• 30 = 2 · 3 · 5
• 8 = 2 · 2 · 2 = 2³
C) If a factor is repeated, take the one with the
highest exponent.
• 2³ has a higherhigher exponent than 2 so, we take 2³2³.
• →→L.C.M. (30, 8) = 3 · 5 ·L.C.M. (30, 8) = 3 · 5 · 2³2³ = 120= 120

11. Another Example:
• Find the L.C.M. of 100100 and 120120.
100 │ 22 120 │ 22
50 │ 22 60 │ 22
25 │ 55 30 │ 22
5 │ 55 15 │ 33
1 │ 5 │ 55
1 │

12. • 100 = 22 · 22 · 55 · 55 = 2²2² · 5²5²
• 120 = 22 · 22 · 22 · 33 · 55 = 2³2³ · 33 · 55
• 2³2³ has a higher exponent than 2²2²..
• 5²5² has a higher exponent than 55..
• →→L.C.M. (100, 120) = 2³L.C.M. (100, 120) = 2³ · 33 · 5² = 6005² = 600

13. Another Example:
• Find the L.C.M. of 120120, 480480 and 180180.
120 │ 22 480 │ 22 180 │ 22
60 │ 22 240 │ 22 90 │ 22
30 │ 22 120 │ 22 45 │ 33
15 │ 33 60 │ 22 15 │ 33
5 │ 55 30 │ 22 5 │ 55
1 │ 15 │ 33 1 │
5 │ 55
1 │

14. • 120 = 2 · 2 · 2 · 3 · 5 = 2³ · 3 · 5
• 480 = 2 · 2 · 2 · 2 · 2 · 3 · 5 = 2⁵ · 3 · 5
• 180 = 2 · 2 · 3 · 3 · 5 = 2² · 3² · 5
• 120 = 2 · 2 · 2 · 3 · 5 = 2³ · 3 · 5
• 480 = 2 · 2 · 2 · 2 · 2 · 3 · 5 = 2⁵2⁵ · 3 · 5
• 180 = 2 · 2 · 3 · 3 · 5 = 2² · 3²3² · 55
• 2⁵2⁵ has ahas a higherhigher exponent thanexponent than 2³ and 2².2³ and 2².
• 3²3² has ahas a higherhigher exponent thanexponent than 3.3.
• →→ L.C.M. (120, 480, 180) = 2⁵ · 3² · 5L.C.M. (120, 480, 180) = 2⁵ · 3² · 5
= 1,440= 1,440