Euclid's Division Lemma: For any positive integers a and b, there exist unique integers q and r such that a = bq + r and 0 ≤ r < b
Fundamental Theorem of Arithmetic: Every positive integer is either a prime number or can be expressed as a product of prime numbers ∈ a unique way
LCM(a, b) × GCD(a, b) = a × b
GCD(a, b) = GCD(b, a mod b)
If a and b are two positive integers, then their HCF is the largest number that divides both a and b
If a and b are two positive integers, then their LCM is the smallest number that is a multiple of both a and b

🪤 The 5 Mistakes That Cost Marks

Not understanding the concept of HCF and LCM and how to calculate them
Not being able to apply Euclid's Division Lemma to find the HCF of two numbers
Not being able to factorize numbers into their prime factors
Not being able to identify the prime factors of a number
Not being able to simplify expressions involving HCF and LCM

✏️ 3 Solved PYQs

Question 1: Find the HCF and LCM of 12 and 15 Step 1: Find the prime factors of 12 and 15 Step 2: Identify the common prime factors and calculate the HCF Step 3: Calculate the LCM using the formula LCM(a, b) = (a × b)/HCF(a, b) Answer: HCF(12, 15) = 3, LCM(12, 15) = 60
Question 2: Find the prime factorization of 84 Step 1: Divide 84 by the smallest prime number, which is 2 Step 2: Continue dividing the quotient by prime numbers until the quotient is 1 Step 3: Write the prime factorization as a product of prime numbers Answer: 84 = 2² × 3 × 7
Question 3: Find the LCM of 24, 30, and 36 Step 1: Find the prime factorization of each number Step 2: Identify the highest power of each prime factor Step 3: Calculate the LCM as the product of the highest powers of each prime factor Answer: LCM (24, 30, 36) = 2³ × 3² × 5 = 360

🧠 The One Thing Most Students Get Wrong

Many students get confused between the concepts of HCF and LCM and are not able to apply them correctly to solve problems
They often struggle to find the prime factorization of numbers and to calculate the HCF and LCM using the prime factorization
It is essential to practice solving problems involving HCF and LCM to become proficient ∈ these concepts

👁️ Ayush's Note

To solve problems involving HCF and LCM, it is crucial to first find the prime factorization of the given numbers
Then, identify the common prime factors to calculate the HCF
Use the formula LCM(a, b) = (a × b)/HCF(a, b) to calculate the LCM
Practice solving problems involving HCF and LCM to become proficient ∈ these concepts

🔁 Last 5 Minutes Box

Review the formulas for HCF and LCM
Practice calculating the HCF and LCM of numbers
Make sure to understand the concept of prime factorization and how to apply it to solve problems
Go through the solved examples and practice problems to reinforce your understanding

📝 Practice MCQs

1. What is the HCF of 18 and 24?

A) 2

B) 3

C) 6

D) 12

Answer: C) 6.

2. What is the LCM of 12 and 15?

A) 30

B) 60

C) 90

D) 120

Answer: B) 60.

3. What is the prime factorization of 48?

A) 2³ × 3

B) 2⁴ × 3

C) 2³ × 3²

D) 2⁴ × 3²

Answer: A) 2³ × 3.

4. What is the HCF of 24, 30, and 36?

A) 2

B) 3

C) 6

D) 12

Answer: B) 6.

5. What is the LCM of 8, 12, and 15?

A) 120

B) 240

C) 360

D) 480

Answer: A) 120.

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

�Let the People Rap�: Cultural Rhetorics Pedagogy and Practices U... — Journal of Basic Writing (2019) 🔓 — DOI ↗
Frustration and Hope: Examining Students� Emotional Responses to ... — Journal of Basic Writing (2019) — DOI ↗
Editors' Column — Journal of Basic Writing (2019) — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Euclid's Division Lemma: For any positive integers a and b, there exist unique integers q and r such that a = bq + r and 0 ≤ r < b
Fundamental Theorem of Arithmetic: Every positive integer is either a prime number or can be expressed as a product of prime numbers ∈ a unique way
LCM(a, b) × GCD(a, b) = a × b
GCD(a, b) = GCD(b, a mod b)
If a and b are two positive integers, then their HCF is the largest number that divides both a and b
If a and b are two positive integers, then their LCM is the smallest number that is a multiple of both a and b

🪤 The 5 Mistakes That Cost Marks

Not understanding the concept of HCF and LCM and how to calculate them
Not being able to apply Euclid's Division Lemma to find the HCF of two numbers
Not being able to factorize numbers into their prime factors
Not being able to identify the prime factors of a number
Not being able to simplify expressions involving HCF and LCM

✏️ 3 Solved PYQs

Question 1: Find the HCF and LCM of 12 and 15 Step 1: Find the prime factors of 12 and 15 Step 2: Identify the common prime factors and calculate the HCF Step 3: Calculate the LCM using the formula LCM(a, b) = (a × b)/HCF(a, b) Answer: HCF(12, 15) = 3, LCM(12, 15) = 60
Question 2: Find the prime factorization of 84 Step 1: Divide 84 by the smallest prime number, which is 2 Step 2: Continue dividing the quotient by prime numbers until the quotient is 1 Step 3: Write the prime factorization as a product of prime numbers Answer: 84 = 2² × 3 × 7
Question 3: Find the LCM of 24, 30, and 36 Step 1: Find the prime factorization of each number Step 2: Identify the highest power of each prime factor Step 3: Calculate the LCM as the product of the highest powers of each prime factor Answer: LCM (24, 30, 36) = 2³ × 3² × 5 = 360

🧠 The One Thing Most Students Get Wrong

Many students get confused between the concepts of HCF and LCM and are not able to apply them correctly to solve problems
They often struggle to find the prime factorization of numbers and to calculate the HCF and LCM using the prime factorization
It is essential to practice solving problems involving HCF and LCM to become proficient ∈ these concepts

👁️ Ayush's Note

To solve problems involving HCF and LCM, it is crucial to first find the prime factorization of the given numbers
Then, identify the common prime factors to calculate the HCF
Use the formula LCM(a, b) = (a × b)/HCF(a, b) to calculate the LCM
Practice solving problems involving HCF and LCM to become proficient ∈ these concepts

🔁 Last 5 Minutes Box

Review the formulas for HCF and LCM
Practice calculating the HCF and LCM of numbers
Make sure to understand the concept of prime factorization and how to apply it to solve problems
Go through the solved examples and practice problems to reinforce your understanding

📝 Practice MCQs

1. What is the HCF of 18 and 24?

A) 2

B) 3

C) 6

D) 12

Answer: C) 6.

2. What is the LCM of 12 and 15?

A) 30

B) 60

C) 90

D) 120

Answer: B) 60.

3. What is the prime factorization of 48?

A) 2³ × 3

B) 2⁴ × 3

C) 2³ × 3²

D) 2⁴ × 3²

Answer: A) 2³ × 3.

4. What is the HCF of 24, 30, and 36?

A) 2

B) 3

C) 6

D) 12

Answer: B) 6.

5. What is the LCM of 8, 12, and 15?

A) 120

B) 240

C) 360

D) 480

Answer: A) 120.

🚀 Ready to Ace Your Exam?

Put your knowledge to the test! Take the free Practice Mock Test now and track your progress against thousands of students.

📚 Academic References

Content verified against peer-reviewed research:

�Let the People Rap�: Cultural Rhetorics Pedagogy and Practices U... — Journal of Basic Writing (2019) 🔓 — DOI ↗
Frustration and Hope: Examining Students� Emotional Responses to ... — Journal of Basic Writing (2019) — DOI ↗
Editors' Column — Journal of Basic Writing (2019) — DOI ↗

🔓 = Open Access article

🎬 Watch video explanations on YouTube →

This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides: