Number system and HCF LCM

Introduction to Number System and HCF & LCM

Numbers are the foundation of mathematics and appear everywhere in daily life-from counting objects to measuring distances. Understanding the number system helps us classify and work with different types of numbers effectively. Alongside this, concepts like factors, multiples, Highest Common Factor (HCF), and Least Common Multiple (LCM) are essential tools that help solve many practical problems such as scheduling, dividing things equally, and simplifying calculations.

In this chapter, we will explore the various types of numbers, learn how to identify factors and multiples, and master the methods to find HCF and LCM. These concepts are not only important for competitive exams but also for building a strong mathematical foundation.

Number System Basics

Let's start by understanding the different types of numbers and how they relate to each other.

Key Concept

Number System Classification

Numbers are grouped into categories based on their properties and uses.

Here is a hierarchical view of the number system:

Natural Numbers (N): These are the counting numbers starting from 1, 2, 3, and so on.

Whole Numbers (W): Natural numbers including zero: 0, 1, 2, 3, ...

Integers (Z): Whole numbers along with their negative counterparts: ..., -3, -2, -1, 0, 1, 2, 3, ...

Rational Numbers (Q): Numbers that can be expressed as a fraction \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q eq 0\). Examples: \(\frac{1}{2}, -\frac{3}{4}, 5\).

Irrational Numbers: Numbers that cannot be expressed as fractions, their decimal expansions are non-terminating and non-repeating. Examples: \(\sqrt{2}, \pi\).

Divisibility Rules

Divisibility rules help us quickly determine if one number is divisible by another without performing full division. This is very useful for factorization and simplifying problems.

Divisible by 2: If the last digit is even (0, 2, 4, 6, 8).
Divisible by 3: If the sum of digits is divisible by 3.
Divisible by 5: If the last digit is 0 or 5.
Divisible by 9: If the sum of digits is divisible by 9.
Divisible by 11: If the difference between the sum of digits in odd positions and even positions is 0 or divisible by 11.

Prime Factorization

A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Examples: 2, 3, 5, 7, 11, 13, ...

A composite number is a natural number greater than 1 that is not prime; it has factors other than 1 and itself.

Prime factorization means expressing a number as a product of its prime factors. This is a key step in finding HCF and LCM.

Let's see how to perform prime factorization using a factor tree:

Explanation: Start with 84. It can be divided into 12 and 7 (since \(12 \times 7 = 84\)). Then break 12 into 2 and 6. Finally, break 6 into 2 and 3. All the leaves of the tree (2, 2, 3, 7) are prime numbers.

So, the prime factorization of 84 is:

\[ 84 = 2 \times 2 \times 3 \times 7 = 2^2 \times 3 \times 7 \]

Finding HCF (Highest Common Factor)

The Highest Common Factor (HCF) of two or more numbers is the largest number that divides all of them exactly (without leaving a remainder). It is also called the Greatest Common Divisor (GCD).

Why is HCF important? It helps in simplifying fractions, dividing things into equal parts, and solving problems involving common factors.

Methods to Find HCF

1. Listing Factors

Write down all factors of each number and find the greatest one common to all.

2. Prime Factorization

Find prime factors of each number, then multiply the common prime factors with the smallest powers.

3. Euclid's Algorithm

This is an efficient method based on the principle that the HCF of two numbers also divides their difference.

graph TD    A[Start with numbers a and b, where a > b]    A --> B[Divide a by b]    B --> C[Find remainder r]    C --> D{Is r = 0?}    D -- No --> E[Set a = b, b = r]    E --> B    D -- Yes --> F[HCF is b]

This process repeats until the remainder is zero. The divisor at that point is the HCF.

Finding LCM (Least Common Multiple)

The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of all of them.

LCM is useful in problems involving synchronization of events, adding fractions, and finding common denominators.

Methods to Find LCM

1. Prime Factorization

Find prime factors of each number. For each prime factor, take the highest power that appears in any number and multiply them.

**Example: Prime Factors of 12 and 18**
Prime Factor	12 = \(2^2 \times 3\)	18 = \(2 \times 3^2\)	Highest Power for LCM
2	2²	2¹	2²
3	3¹	3²	3²

So, \(\text{LCM} = 2^2 \times 3^2 = 4 \times 9 = 36\).

2. Using HCF

There is a useful relationship between HCF and LCM:

Relationship between HCF and LCM

\[HCF(a,b) \times LCM(a,b) = a \times b\]

Used to find LCM if HCF and numbers are known, or vice versa

a, b = two given numbers

HCF(a,b) = highest common factor

LCM(a,b) = least common multiple

Worked Examples

Example 1: Finding HCF and LCM of 36 and 48 Easy

Find the HCF and LCM of 36 and 48 using prime factorization.

Step 1: Find prime factors of 36.

36 = 2 x 2 x 3 x 3 = \(2^2 \times 3^2\)

Step 2: Find prime factors of 48.

48 = 2 x 2 x 2 x 2 x 3 = \(2^4 \times 3^1\)

Step 3: For HCF, take the lowest powers of common primes.

HCF = \(2^{\min(2,4)} \times 3^{\min(2,1)} = 2^2 \times 3^1 = 4 \times 3 = 12\)

Step 4: For LCM, take the highest powers of all primes.

LCM = \(2^{\max(2,4)} \times 3^{\max(2,1)} = 2^4 \times 3^2 = 16 \times 9 = 144\)

Answer: HCF = 12, LCM = 144

Example 2: Using Euclid's Algorithm to find HCF of 270 and 192 Medium

Find the HCF of 270 and 192 using Euclid's algorithm.

Step 1: Divide 270 by 192.

270 / 192 = 1 remainder 78 (since 270 = 192 x 1 + 78)

Step 2: Now divide 192 by 78.

192 / 78 = 2 remainder 36 (192 = 78 x 2 + 36)

Step 3: Divide 78 by 36.

78 / 36 = 2 remainder 6 (78 = 36 x 2 + 6)

Step 4: Divide 36 by 6.

36 / 6 = 6 remainder 0 (36 = 6 x 6 + 0)

Step 5: Since remainder is 0, HCF is the divisor 6.

Answer: HCF(270, 192) = 6

Example 3: Word Problem - Scheduling Events Using LCM Medium

Two buses leave a station at the same time. One bus returns every 12 minutes, and the other every 18 minutes. After how many minutes will both buses return together again?

Step 1: Identify the intervals: 12 minutes and 18 minutes.

Step 2: Find LCM of 12 and 18.

Prime factors:

12 = \(2^2 \times 3\)
18 = \(2 \times 3^2\)

LCM = \(2^{\max(2,1)} \times 3^{\max(1,2)} = 2^2 \times 3^2 = 4 \times 9 = 36\)

Answer: Both buses will return together after 36 minutes.

Example 4: Finding LCM Using HCF for 24 and 90 Easy

Given that HCF of 24 and 90 is 6, find their LCM.

Step 1: Use the formula:

\[ \text{HCF} \times \text{LCM} = \text{Product of numbers} \]

Step 2: Substitute the values:

\[ 6 \times \text{LCM} = 24 \times 90 \]

\[ 6 \times \text{LCM} = 2160 \]

Step 3: Solve for LCM:

\[ \text{LCM} = \frac{2160}{6} = 360 \]

Answer: LCM of 24 and 90 is 360.

Example 5: Complex Problem Involving HCF, LCM and Multiple Numbers Hard

Three numbers have an HCF of 5 and an LCM of 600. If two of the numbers are 25 and 60, find the third number.

Step 1: Let the third number be \(x\).

Step 2: Since HCF is 5, express numbers as multiples of 5:

25 = 5 x 5, 60 = 5 x 12, \(x = 5 \times k\)

Step 3: Find LCM of the three numbers:

LCM(25, 60, \(x\)) = 600

Divide all numbers by 5 to simplify:

5, 12, \(k\)

LCM(5, 12, \(k\)) = \(\frac{600}{5} = 120\)

Step 4: Find LCM of 5 and 12:

Prime factors:

5 = 5
12 = \(2^2 \times 3\)

LCM(5, 12) = \(2^2 \times 3 \times 5 = 60\)

Step 5: Since LCM(5,12,k) = 120, and LCM(5,12) = 60, \(k\) must introduce a factor to increase LCM to 120.

120 / 60 = 2, so \(k\) must have factor 2.

Step 6: \(k\) must be a multiple of 2 but not introduce factors that reduce HCF.

Try \(k = 2\).

Step 7: Third number \(x = 5 \times 2 = 10\).

Answer: The third number is 10.

Tips & Tricks

Tip: Use prime factorization to find HCF and LCM quickly by comparing powers of prime factors.

When to use: When numbers are not too large and factorization is straightforward.

Tip: Apply Euclid's algorithm for HCF to save time on large numbers instead of listing factors.

When to use: For large numbers where factorization is time-consuming.

Tip: Remember the formula \( \text{HCF} \times \text{LCM} = \text{Product of numbers} \) to find one if the other is known.

When to use: When either HCF or LCM is missing but the other and the numbers are known.

Tip: For word problems involving schedules or repeated events, use LCM to find the common occurrence time.

When to use: Problems involving periodic events or cycles.

Tip: Check divisibility rules first to quickly identify factors and reduce calculation time.

When to use: At the start of factorization or when simplifying numbers.

Common Mistakes to Avoid

❌ Confusing HCF with LCM or mixing their calculation methods.

✓ Remember HCF is the greatest factor common to numbers; LCM is the smallest multiple common to numbers.

Why: Students often confuse the two because both involve factors and multiples.

❌ Not fully breaking down numbers into prime factors leading to incorrect HCF or LCM.

✓ Always perform complete prime factorization before calculating HCF or LCM.

Why: Partial factorization misses some prime factors, causing errors.

❌ Using addition or subtraction instead of multiplication in the HCF x LCM = product formula.

✓ Use multiplication, not addition or subtraction.

Why: Misremembering the formula leads to wrong answers.

❌ Forgetting to check if numbers are divisible by smaller primes first.

✓ Apply divisibility rules before attempting factorization.

Why: Skipping this step makes factorization longer and error-prone.

❌ Applying Euclid's algorithm incorrectly by mixing up dividend and divisor.

✓ Always divide the larger number by the smaller and use remainder correctly.

Why: Incorrect order disrupts the algorithm's logic.

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Number system and HCF LCM

Introduction to Number System and HCF & LCM

Number System Basics

Number System Classification

Divisibility Rules

Prime Factorization

Finding HCF (Highest Common Factor)

Methods to Find HCF

1. Listing Factors

2. Prime Factorization

3. Euclid's Algorithm

Finding LCM (Least Common Multiple)

Methods to Find LCM

1. Prime Factorization

2. Using HCF

Relationship between HCF and LCM

Worked Examples

Tips & Tricks

Common Mistakes to Avoid

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