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In mathematics, understanding how numbers relate to each other through their factors and multiples is fundamental. Two important concepts that arise from these relationships are the Highest Common Factor (HCF) and the Least Common Multiple (LCM). These concepts help us solve many practical problems, such as dividing items into equal groups, scheduling events, and sharing costs fairly.
Before diving into HCF and LCM, we will first explore what factors and multiples are, then learn how to find HCF and LCM using various methods. Finally, we will apply these concepts to solve real-life problems, preparing you well for entrance exams like the Meghalaya Police SI written test.
Factors of a number are the numbers that divide it exactly without leaving a remainder. For example, the factors of 12 are numbers that divide 12 completely.
Multiples of a number are the numbers obtained by multiplying that number by integers. For example, multiples of 4 are numbers like 4, 8, 12, 16, and so on.
Let's look at factors and multiples with examples:
Example: Factors of 12 are 1, 2, 3, 4, 6, and 12 because these numbers divide 12 exactly.
Multiples of 4 are 4, 8, 12, 16, 20, 24, and so on, because these numbers are obtained by multiplying 4 by 1, 2, 3, 4, 5, 6, etc.
Prime factorization is the process of expressing a number as a product of its prime factors. A prime number is a number greater than 1 that has no factors other than 1 and itself.
Prime factorization helps us find the HCF and LCM of numbers by breaking them down into their basic building blocks.
We use a factor tree to find prime factors. Let's see how it works with the number 60:
So, the prime factors of 60 are 2, 2, 3, and 5 because:
60 = 6 x 10 = (2 x 3) x (2 x 5) = 2 x 2 x 3 x 5
Now that we know factors and multiples, let's define the two key concepts:
Let's compare factors and multiples of some numbers to understand this better:
| Number | Factors | Multiples (first 5) |
|---|---|---|
| 12 | 1, 2, 3, 4, 6, 12 | 12, 24, 36, 48, 60 |
| 18 | 1, 2, 3, 6, 9, 18 | 18, 36, 54, 72, 90 |
| 4 | 1, 2, 4 | 4, 8, 12, 16, 20 |
| 6 | 1, 2, 3, 6 | 6, 12, 18, 24, 30 |
For example, the common factors of 12 and 18 are 1, 2, 3, and 6. The highest among these is 6, so HCF(12, 18) = 6.
The common multiples of 4 and 6 are 12, 24, 36, ... The least among these is 12, so LCM(4, 6) = 12.
There are several methods to find the HCF and LCM of numbers. We will learn three popular ones:
Prime Factorization Method: Break each number into prime factors, then:
Division Method: Divide the numbers by common prime numbers step-by-step until no further division is possible.
graph TD Start[Start with given numbers] Divide[Divide by common prime divisor] Record[Record divisor] Update[Update quotient numbers] Check[Can numbers be divided further?] End[Multiply recorded divisors for HCF and LCM] Start --> Divide Divide --> Record Record --> Update Update --> Check Check -- Yes --> Divide Check -- No --> End
Using Venn Diagrams: Draw circles representing prime factors of each number. The intersection shows common factors (for HCF), and the union shows all factors (for LCM).
Step 1: Find prime factors of 36.
36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3
Prime factors of 36: \(2^2 \times 3^2\)
Step 2: Find prime factors of 48.
48 = 2 x 24 = 2 x 2 x 12 = 2 x 2 x 2 x 6 = 2 x 2 x 2 x 2 x 3
Prime factors of 48: \(2^4 \times 3^1\)
Step 3: Find HCF by taking common prime factors with the lowest powers.
Common prime factors: 2 and 3
Lowest powers: \(2^2\) and \(3^1\)
HCF = \(2^2 \times 3^1 = 4 \times 3 = 12\)
Step 4: Find LCM by taking all prime factors with the highest powers.
Highest powers: \(2^4\) and \(3^2\)
LCM = \(2^4 \times 3^2 = 16 \times 9 = 144\)
Answer: HCF(36, 48) = 12 and LCM(36, 48) = 144
Step 1: Identify the problem as finding the LCM of 12 and 15.
Step 2: Find prime factors:
12 = \(2^2 \times 3\)
15 = \(3 \times 5\)
Step 3: Take highest powers for LCM:
LCM = \(2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\)
Answer: The two events will coincide again after 60 days.
Step 1: This is a problem of finding the HCF of 72 and 96.
Step 2: Prime factorization:
72 = \(2^3 \times 3^2\)
96 = \(2^5 \times 3^1\)
Step 3: HCF = common prime factors with lowest powers:
HCF = \(2^3 \times 3^1 = 8 \times 3 = 24\)
Step 4: Number of items per pack = 24
Number of packs = \(\frac{72}{24} = 3\) packs of pens and \(\frac{96}{24} = 4\) packs of pencils
Answer: Each pack will have 24 items. There will be 3 packs of pens and 4 packs of pencils.
Step 1: Use the relationship:
\[ \text{HCF} \times \text{LCM} = \text{Product of the two numbers} \]
Step 2: Substitute known values:
6 x 180 = 30 x (Other number)
1080 = 30 x (Other number)
Step 3: Find the other number:
Other number = \(\frac{1080}{30} = 36\)
Answer: The other number is 36.
Step 1: This is a problem of finding the LCM of 20 and 30.
Step 2: Prime factorization:
20 = \(2^2 \times 5\)
30 = \(2 \times 3 \times 5\)
Step 3: LCM = highest powers of all prime factors:
LCM = \(2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60\)
Answer: They will pay together again after 60 days.
When to use: When numbers are manageable and can be factored easily.
When to use: To find missing values when two of the three are known.
When to use: To understand and solve problems involving multiple numbers.
When to use: When interpreting real-life application problems.
When to use: To speed up factorization and avoid errors.
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