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Factors and multiples (LCM, HCF/GCD) — KCSE Mathematics

KCSE Mathematics · 101 practice questions · 4 syllabus objectives · 4 revision lessons

37 easy34 medium30 hard

Last updated · Aligned to the KNEC KCSE syllabus

What You'll Learn

Key learning outcomes for this topic, aligned to the KNEC KCSE syllabus.

Find the prime factorisation of a number and use it to determine all factors of the number

Calculate the Lowest Common Multiple (LCM) and Highest Common Factor (HCF/GCD) of two or more numbers using prime factorisation

Apply LCM and HCF to solve word problems involving repeated events, sharing and grouping

Factors and multiples (LCM, HCF/GCD)

Revision Notes

Concise lesson notes for Factors and multiples (LCM, HCF/GCD), written to the KCSE Mathematics marking standard. Read the first lesson free below.

Finding Prime Factorisation and Factors

To find the prime factorisation of a number, we express it as a product of prime numbers. This method helps us determine all factors of the number. Steps to find prime factorisation:

  1. Start with the smallest prime number (2) and divide the number.
  2. Continue dividing by prime numbers until you reach 1.
  3. Write the number as a product of its prime factors.
  4. Use the prime factors to find all factors by considering different combinations.

Example: Find the prime factorisation of 60.

  • Divide by 2: 60 ÷ 2 = 30
  • Divide by 2 again: 30 ÷ 2 = 15
  • Divide by 3: 15 ÷ 3 = 5
  • Finally, 5 is a prime number.
    Thus, the prime factorisation of 60 is:
    60 = 2² × 3¹ × 5¹
    To find all factors:
  • Use the exponents in the prime factorisation.
  • Factors are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Key points to remember

  • Prime factorisation expresses a number as prime factors.
  • Use division by prime numbers to find prime factors.
  • All factors can be determined from prime factorisation.
  • Exponents in prime factorisation help find combinations.

Worked example

Find the prime factorisation of 28 and all its factors.

  • 28 = 2² × 7¹.
  • Factors are: 1, 2, 4, 7, 14, 28.

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Lesson 2: Finding LCM and HCF using Prime Factorisation

Objective: Calculate the Lowest Common Multiple (LCM) and Highest Common Factor (HCF/GCD) of two or more numbers using prime factorisation

To calculate the Lowest Common Multiple (LCM) and Highest Common Factor (HCF) using prime factorisation, follow these steps:

  1. Prime Factorisation: Break down each number into its prime factors.

    • Example: For 12 and 18:
      • 12 = 2² × 3¹
      • 18 = 2¹ × 3²

  2. Finding HCF: Identify the common prime factors and take the lowest power.

    • Common factors: 2¹ and 3¹
    • HCF = 2¹ × 3¹ = 6

  3. Finding LCM: Identify all prime factors and take the highest power.

    • All factors: 2² and 3²
    • LCM = 2² × 3² = 36

In summary, the HCF of 12 and 18 is 6, while the LCM is 36. Remember, the HCF is the product of the lowest powers of common factors, and the LCM is the product of the highest powers of all factors.

  • Use prime factorisation for each number.
  • HCF is the product of lowest powers of common factors.
  • LCM is the product of highest powers of all factors.

Calculate the LCM and HCF of 24 and 36.

  • 24 = 2³ × 3¹
  • 36 = 2² × 3²
  • HCF = 2² × 3¹ = 12
  • LCM = 2³ × 3² = 72.
Lesson 3: Applying LCM and HCF in Real-Life Problems

Objective: Apply LCM and HCF to solve word problems involving repeated events, sharing and grouping

To solve word problems involving repeated events, sharing, and grouping, we utilize the Lowest Common Multiple (LCM) and Highest Common Factor (HCF).

LCM is used when we need to find a common time or interval for events that repeat. For example, if two bells ring every 12 minutes and 15 minutes, we find the LCM to determine when they will ring together again.

HCF is useful when sharing items among groups. For instance, if you have 24 apples and 36 oranges, the HCF helps you find the largest number of equal groups you can create without leftovers.

Steps to solve problems:

  1. Identify the numbers involved in the problem.
  2. Calculate the LCM or HCF as required.
  3. Apply the results to answer the question.

Example: If two friends have 18 and 24 candies respectively, how many candies can they share equally?

  • Find HCF of 18 and 24: HCF = 6.
  • They can share the candies in groups of 6.
  • LCM is used for repeated events and timing problems.
  • HCF is applied in sharing and grouping items equally.
  • Identify numbers clearly before calculating LCM or HCF.
  • Use LCM to find when events coincide again.
  • Use HCF to determine maximum equal groups.

If two trains leave a station every 30 minutes and 45 minutes, when will they next leave together?

  • LCM of 30 and 45 is 90 minutes. They will leave together again in 90 minutes.
Lesson 4: Understanding LCM and HCF/GCD

Objective: Factors and multiples (LCM, HCF/GCD)

In mathematics, the Least Common Multiple (LCM) and Highest Common Factor (HCF), also known as the Greatest Common Divisor (GCD), are essential concepts for working with factors and multiples.

  • LCM is the smallest multiple that two or more numbers share. To find the LCM, list the multiples of each number and identify the smallest common one.
  • HCF/GCD is the largest number that divides two or more numbers without leaving a remainder. To find the HCF, list the factors of each number and identify the largest common factor.

For example, to find the LCM and HCF of 12 and 18:

  • Multiples of 12: 12, 24, 36, 48, 60...
  • Multiples of 18: 18, 36, 54, 72...
  • The LCM is 36 (smallest common multiple).
  • Factors of 12: 1, 2, 3, 4, 6, 12
  • Factors of 18: 1, 2, 3, 6, 9, 18
  • The HCF is 6 (largest common factor).
  • LCM is the smallest common multiple of numbers.
  • HCF/GCD is the largest common divisor of numbers.
  • List multiples for LCM; list factors for HCF.
  • Use prime factorization for efficient calculations.
  • Understanding these concepts aids in problem-solving.

Find the LCM and HCF of 8 and 12.

  • LCM: Multiples of 8 (8, 16, 24, 32) and 12 (12, 24, 36) → LCM is 24.
  • HCF: Factors of 8 (1, 2, 4, 8) and 12 (1, 2, 3, 4, 6, 12) → HCF is 4.

Sample Questions

Read 3 questions and answers free. Sign up to access all 101 questions with full KNEC-style marking schemes and a personalised study plan.

1
easySHORT ANSWER2 marks

State the largest number of crates that can be evenly divided into groups of 12 and 16. (2 marks)

Answer & marking scheme

Part (a) — 2 marks
Prime factorise 12 and 16 (1 mk)
State the HCF as 4 (1 mk)
2
easySHORT ANSWER3 marks

State the least number of bicycles that can be arranged in rows of 6, 8, or 10 without any remaining bicycles. (3 marks)

Answer & marking scheme

Part (a) — 3 marks
Prime factorise 6, 8, and 10 (1 mk)
Identify the highest powers of each prime factor (1 mk)
State the LCM as 120 (1 mk)
3
easySHORT ANSWER4 marks

Using prime factorisation, calculate the HCF and LCM of the numbers 18 and 30. (4 marks)

Answer & marking scheme

Part (a) — 2 marks
HCF = 6 (2 mks)
Part (b) — 2 marks
LCM = 90 (2 mks)
4

Name the Highest Common Factor (HCF) and Lowest Common Multiple (LCM) of the numbers 24 and 36 using prime factorisation. (4 marks)

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Frequently asked questions

What does the KCSE Mathematics topic "Factors and multiples (LCM, HCF/GCD)" cover?

Factors and multiples (LCM, HCF/GCD) covers Find the prime factorisation of a number and use it to determine all factors of the number; Calculate the Lowest Common Multiple (LCM) and Highest Common Factor (HCF/GCD) of two or more numbers using prime factorisation; Apply LCM and HCF to solve word problems involving repeated events, sharing and grouping, and more, all aligned to the official KNEC KCSE Mathematics syllabus.

How many practice questions are available for Factors and multiples (LCM, HCF/GCD)?

HighMarks has 101 Factors and multiples (LCM, HCF/GCD) practice questions for KCSE Mathematics, each with a full marking scheme. The first 3 are free; sign up to access the rest, plus all KCSE mock exams and past papers.

Are these aligned with the KNEC KCSE syllabus?

Yes. Every objective on this page is taken directly from the official KNEC KCSE Mathematics syllabus. Practice questions match the KCSE exam format and are graded against the standard KNEC marking scheme.

How should I revise Factors and multiples (LCM, HCF/GCD) for the KCSE exam?

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