L.C.M and H.C.F Basic 6 Mathematics Lesson Note

Lowest Common Multiple (L.C.M)

What is L.C.M?

The L.C.M of two or more numbers is the smallest number that is a multiple of all of them.

Method 1: Listing Method

Find L.C.M of 12 and 18:

Multiples of 12: 12, 24, 36, 48, 60, 72… Multiples of 18: 18, 36, 54, 72, 90…

Common multiples: 36, 72… L.C.M = 36 (smallest common multiple)

Method 2: Prime Factorization

Find L.C.M of 24 and 36:

24 = 2³ × 3¹ 36 = 2² × 3²

L.C.M = 2³ × 3² = 8 × 9 = 72

Method 3: Division Method

Find L.C.M of 15, 20, 25:

2 | 15, 20, 25

3 | 15, 10, 25

5 | 5,  10, 25

  | 1,   2,  5

2 | 1,   2,  5

  | 1,   1,  5

5 | 1,   1,  5

  | 1,   1,  1

L.C.M = 2 × 3 × 5 × 2 × 5 = 300

Exercise A – Find L.C.M

Find the L.C.M of these numbers:

1. 6 and 8 = ______
2. 12 and 15 = ______
3. 20 and 30 = ______
4. 14, 21, 28 = ______
5. 18, 24, 36 = ______

Highest Common Factor (H.C.F)

What is H.C.F?

The H.C.F of two or more numbers is the largest number that divides all of them exactly.

Method 1: Listing Factors

Find H.C.F of 18 and 24:

Factors of 18: 1, 2, 3, 6, 9, 18 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Common factors: 1, 2, 3, 6 H.C.F = 6 (largest common factor)

Method 2: Prime Factorization

Find H.C.F of 60 and 84:

60 = 2² × 3¹ × 5¹ 84 = 2² × 3¹ × 7¹

H.C.F = 2² × 3¹ = 4 × 3 = 12

Method 3: Division Method

Find H.C.F of 48 and 72:

72 ÷ 48 = 1 remainder 24

48 ÷ 24 = 2 remainder 0

H.C.F = 24 (last non-zero remainder)

Exercise B – Find H.C.F

Find the H.C.F of these numbers:

1. 12 and 18 = ______
2. 24 and 36 = ______
3. 45 and 60 = ______
4. 28, 42, 56 = ______
5. 36, 48, 60 = ______

Exercise C – Mixed L.C.M and H.C.F

Find both L.C.M and H.C.F:

1. Numbers: 16 and 24 L.C.M = ______ H.C.F = ______
2. Numbers: 30 and 45 L.C.M = ______ H.C.F = ______
3. Numbers: 21, 35, 49 L.C.M = ______ H.C.F = ______
4. Numbers: 72 and 108 L.C.M = ______ H.C.F = ______

Real Life Problems on L.C.M and H.C.F

L.C.M Problems

Example: Three bells ring at intervals of 6, 8, and 12 minutes. If they ring together at 9:00 AM, when will they ring together again?

Solution: L.C.M of 6, 8, 12 = 24 minutes They will ring together again at 9:24 AM

H.C.F Problems

Example: A rectangular plot is 72m by 48m. What is the length of the largest square tiles that can be used to tile the plot exactly?

Solution: H.C.F of 72 and 48 = 24m The largest square tile is 24m × 24m

Exercise D – L.C.M Word Problems

Solve these problems:

1. Traffic Lights: Three traffic lights change every 20, 30, and 45 seconds. If they change together at 8:00 AM, when will they change together again? Answer: ______
2. Bus Schedule: Bus A comes every 15 minutes, Bus B every 20 minutes, and Bus C every 25 minutes. If they all arrive at 7:00 AM, when will they arrive together next? Answer: ______
3. School Assembly: Primary 4 has assembly every 6 days, Primary 5 every 8 days, and Primary 6 every 12 days. If they have assembly together today, after how many days will they have assembly together again? Answer: ______ days

Exercise E – H.C.F Word Problems

Solve these problems:

1. Gift Sharing: Kemi has 48 oranges and 72 apples to share equally among children. What is the maximum number of children that can receive the same number of each fruit? Answer: ______
2. Classroom Arrangement: A teacher wants to arrange 84 boys and 126 girls into equal groups with the same number of boys and girls in each group. What is the maximum number of groups possible? Answer: ______
3. Fabric Cutting: Two pieces of cloth measuring 96cm and 144cm need to be cut into equal squares. What is the side length of the largest possible square? Answer: ______ cm

Exercise F – Mixed Real Problems

Solve these problems using L.C.M or H.C.F:

1. Sports Day: Students are arranged in rows. When arranged in rows of 12, 15, or 18, there are always 3 students left over. What is the smallest possible number of students? Answer: ______
2. Book Distribution: A library has 156 English books and 234 Mathematics books. They want to pack them in boxes such that each box has the same number of English books and the same number of Mathematics books. What is the maximum number of boxes needed? Answer: ______
3. Delivery Schedule: A shop receives bread every 4 days, milk every 6 days, and eggs every 8 days. If all three items are delivered today, after how many days will all three be delivered together again? Answer: ______ days

Quantitative Reasoning

Relationship Between L.C.M and H.C.F

Important Formula: For any two numbers a and b: L.C.M × H.C.F = a × b

Example: If two numbers are 12 and 18: L.C.M = 36, H.C.F = 6 Check: 36 × 6 = 216 and 12 × 18 = 216 ✓

Exercise G – Using the Formula

Use the relationship formula to find missing values:

1. Two numbers are 20 and 30 L.C.M = ______, H.C.F = ______ Check: L.C.M × H.C.F = ______
2. L.C.M = 72, H.C.F = 8, one number = 24 Find the other number: ______
3. Two numbers are 45 and 60 Verify: L.C.M × H.C.F = 45 × 60 L.C.M = ______, H.C.F = ______

Exercise H – Patterns and Logic

Find the pattern and solve:

1. L.C.M Pattern: L.C.M of (2,3) = 6 L.C.M of (3,4) = 12 L.C.M of (4,5) = ______ L.C.M of (5,6) = ______
2. H.C.F Pattern: H.C.F of (12,18) = 6 H.C.F of (24,36) = 12 H.C.F of (36,54) = ______ H.C.F of (48,72) = ______

Exercise I – Mental Calculations

Find these quickly using number properties:

1. L.C.M of 7 and 11 = ______ (both are prime)
2. H.C.F of 13 and 17 = ______ (both are prime)
3. L.C.M of 8 and 12 = ______ (find quickly)
4. H.C.F of 100 and 150 = ______ (think of common factors)

Exercise J – Problem Solving

Solve these reasoning problems:

1. Number Puzzle: Two numbers have L.C.M = 60 and H.C.F = 4. If one number is 12, what is the other number? Answer: ______
2. Logic Problem: The L.C.M of two numbers is 6 times their H.C.F. If H.C.F = 8, what is the L.C.M? Answer: ______
3. Application Problem: In a factory, Machine A completes a cycle every 18 minutes and Machine B every 24 minutes. If they start together, after how long will they both complete their cycles at the same time? Answer: ______ minutes

Exercise K – Advanced Reasoning

Think critically and solve:

1. Optimization Problem: A rectangular garden is 84m by 126m. It needs to be divided into identical square plots. What is the side length of the largest possible square plot, and how many such plots will there be? Answer: Side length: ______ m, Number of plots: ______
2. Time Management: Three friends meet every 12, 15, and 20 days respectively. If they meet today, what is the probability they will meet again within 50 days? Answer: ______
3. Resource Planning: A school has 168 boys and 252 girls. For a sports event, they want to form teams with equal numbers of boys and equal numbers of girls. What is the maximum number of teams possible, and how many boys and girls will be in each team? Answer: Teams: ______, Boys per team: ______, Girls per team: ______

Answer Key

Exercise A:

1. 24
2. 60
3. 60
4. 84
5. 72

Exercise B:

1. 6
2. 12
3. 15
4. 14
5. 12

Exercise C:

1. L.C.M = 48, H.C.F = 8
2. L.C.M = 90, H.C.F = 15
3. L.C.M = 147, H.C.F = 7
4. L.C.M = 216, H.C.F = 36

Exercise D:

1. After 180 seconds (3 minutes)
2. After 300 minutes (5 hours)
3. 24 days

Exercise E:

1. 24 children
2. 42 groups
3. 48 cm

Exercise F:

1. 183 students
2. 39 boxes
3. 24 days

Exercise G:

1. L.C.M = 60, H.C.F = 10, Check = 600
2. Other number = 24
3. L.C.M = 180, H.C.F = 15

Exercise H:

1. 20, 30
2. 18, 24

Exercise I:

1. 77
2. 1
3. 24
4. 50

Exercise J:

1. 20
2. 48
3. 72 minutes

Exercise K:

1. 42 m, 6 plots
2. Yes (they meet after 60 days)
3. 84 teams, 2 boys, 3 girls