Math 6 Chapter 1 Lesson 18: Least common multiple
1. Summary of theory Tóm
1.1. Common multiples
Example 1: Note that the numbers 0, 6, 12, 18, etc. are both multiples of 3 and multiples of 6, then we say “they are common multiples of 3 and 6”.
From there, we have the definition:
Given two numbers a and b. If there is a number d satisfying:
\(d\,\, \vdots \,\,a\) and \(d\,\, \vdots \,\,b\)
then d is called a common multiple of a and b.
The set of multiples of two numbers a and b is denoted by BC(a, b).
Attention:
We need to pay attention to:
* If \(x \in BC(a,b,c,…)\) then \(x\,\, \vdots \,\,a,\,x\,\, \vdots \,\, b,\,x\,\, \vdots \,\,c,…\)
* \(BC(a,b) = B(a)\,\, \cap \,\,B(b)\)
1.2. Least common multiple
Example 2: We have
B(6) = {0, 6, 12, 18, 24, 30,…}
B(8) = {0, 8, 16, 24, 32, 45,…}
\( \Rightarrow \) BC(6, 8) = {0, 24, 48,…}
then, the smallest nonzero number in the set BC(6, 8) is 24. We say 24 is the least common multiple of 6 and 8.
From there, we have define:
The least common multiple of a, b is the smallest nonzero number in the set of common multiples of a and b. The symbol BCNN(a, b).
Comment:
* BCNN(a, 1) = a.
* BCNN(a, b, 1) = BCNN(a, b).
* All common multiples of a and b are BCNN(a, b).
1.3. How to find BCNN
Problem: Find the state balance sheet(a, b, c, …)
Solution method
We do the following three steps:
Step 1: Factor the numbers into prime factors.
Step 2: Pick out the common and special factors.
Step 3: Calculate the selected factors, each factor taken with its largest exponent. That product is the BCNN that must be found.
Attention:
We can find BCNN by the following calculation:
UCLN(a, b) . BCNN(a,b) = ab
Example 3: Let’s define:
a. State Report (8,18,28)
b. State Report(9, 26)
c. State Report(150, 25, 75)
Solution
We in turn do:
* Parsing numbers into prime factors:
\(\begin{array}{l}8 = {2^3}\\18 = {2.3^2}\\28 = {2^2}.7\end{array}\)
Pick out the common and unique prime factors: 2, 3, 7.
The factor 2 has the largest exponent of 3, 3 has the largest exponent of 2, and 7 has the largest exponent of 1.
Then:
\(BCNN\left( {8,{\rm{ }}18,{\rm{ }}28} \right) = {2^3}{.3^2}.7 = 504\)
b. Comment that:
GCC(8, 19) = 1
Therefore, infer:
State balance sheet(9, 26) = 9 . 26 = 243.
c. Comment that:
\(\begin{array}{l}150\,\,\, \vdots \,\,\,25\\150\,\,\, \vdots \,\,\,75\end{array}\ )
Therefore, infer:
State balance sheet(150, 25, 75) = 150
Attention:
We need to pay attention to:
* If (a, b) = 1, then BCNN(a, b) = ab
* If \(a \vdots b\) and \(a \vdots c\) then BCNN(a,b,c,…)=a.
* To find the common multiple of the given numbers, we can find the multiples of the BCNN of those numbers.
2. Illustrated exercise
Question 1: Find the smallest natural number such that when divided by 3, the remainder is 2, when divided by 7, the remainder is 6, and when divided by 25, the remainder is 24.
Solution guide:
Suppose a is the number to find.
Since a is left remainder 2 when divided by 3, remainder 6 when divided by 7 and remainder 24 when divided by 25, a + 1 is divisible by 2, 7, 25.
Therefore
a = State balance sheet(3, 7, 25)
We have:
BCNN(3,7,25) \({3.5^2} = 7 = 525\)
So the number to find a = 254.
Verse 2: There are three square boxes: the red box is 8cm high, the blue box is 7cm high, and the yellow box is 12cm high. People are arranged into three equal piles, each stack has a color. Find the minimum height of the stack of boxes.
Solution guide:
Assume the minimum height of each stack is a (cm)
We have:
a = BCNN(7, 8, 12) = \({2^3}.3.7 = 168\) (cm)
So the minimum height of the stack of boxes is 168cm.
Question 3: Find the natural number a. Know that the number is divisible by 7 and when divided by 2, by 3, by 4, by 5, by 6 all have remainder 1 and a < 400.
Solution guide:
We have:
a – 1 = BC(2, 3, 4, 5, 6)
\(\begin{array}{l} \Rightarrow a – 1 \in {\rm{\{ }}60,120,180,240,300,360\} \\ \Rightarrow a \in {\rm{\{ }}61,121,181,241,301,361\} \end{array }\)
Since \(a \vdots 7\) so a = 301
So a = 301
3. Practice
3.1. Essay exercises
Question 1: Find the balance sheet of:
a) 20 and 35
b) 12; 25 and 26
Verse 2: Find the common multiples less than 500 of 30 and 45.
Question 1: Find the smallest natural number such that when divided by 5 the remainder 4, when divided by 9 the remainder 8, when divided by 13, the remainder 12.
3.2. Multiple choice exercises Bài
Question 1: Find state reports (9; 10; 11)
A. 90
B. 99
C. 110
D. 990
Verse 2: Find the common multiple of 15 and 25 that is less than 400
A. 0, 75, 150, 225, 300, 375
B. 0, 75, 150, 225, 300
C. 75, 150, 225, 300, 375
D. 0, 75, 225, 300, 375
Question 3: Choose the false statement:
A. To find the common multiples of given numbers, we can find multiples of the BCNN of those numbers
B. The least common multiple of two or more numbers is the smallest nonzero number in the set of common multiples of those numbers.
C. All natural numbers are multiples of 1
D. If a is divisible by m and a is divisible by n, then a is not divisible by the BCNN of m and n
Question 4: Some books when arranged in bundles of 10 books, 12 books, 15 books, 18 books are just enough bundles. Know the number of books between 200 and 500. Calculate the number of books.
A. 240
B. 300
C. 360
D. 540
Question 5: Find the threedigit common multiples of 63, 35 and 105
A. 315, 630, 945
B. 630, 945, 1260
C. 630, 945
D. 315, 630
4. Conclusion
Through this Least Common Multiple lesson, you need to complete some of the objectives given by the lesson, such as:

The concept of the common multiple, the least common multiple.

How to find BCNN of numbers.

How to find common multiples by finding BCNN.
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