📘 Notes on LCM (Least
Common Multiple) and HCF (Highest Common Factor)
🔹 Definitions:
LCM (Least Common Multiple):
The smallest number which is exactly divisible by each of the given numbers.
✅
Use case: Problems involving synchronization, repetition of events.
HCF (Highest Common Factor):
The greatest number that divides each of the given numbers exactly (without
leaving a remainder).
✅
Use case: Simplifying fractions, dividing quantities equally.
🔹 Methods to Find LCM and
HCF:
1. Prime Factorization Method
- Break
each number into its prime factors.
- LCM:
Take the highest power of all prime numbers.
- HCF:
Take the lowest power of only the common prime numbers.
Example:
Find the LCM and HCF of 12 and 18.
- 12 =
2² × 3
- 18 =
2 × 3²
👉 LCM = 2² × 3² = 36
👉
HCF = 2 × 3 = 6
2. Division Method (for HCF only)
- Divide
the larger number by the smaller.
- Replace
larger with the smaller and smaller with the remainder.
- Repeat
until remainder is 0.
The divisor at this stage is the HCF.
Example: HCF of 48 and 18
48 ÷ 18 = 2 (rem 12)
18 ÷ 12 = 1 (rem 6)
12 ÷ 6 = 2 (rem 0)
👉
HCF = 6
3. Using the Relation Between LCM and HCF:
LCM×HCF=Product of the numbers\text{LCM}
\times \text{HCF} = \text{Product of the
numbers}LCM×HCF=Product of the numbers
Example: If LCM = 60, HCF = 6 for two numbers,
then the product = 60 × 6 = 360.
🔹 Important Concepts:
🔸 Co-prime Numbers:
Two numbers are co-prime if their HCF is 1.
👉
Example: 4 and 9
🔸 LCM of Fractions:
LCM of fractions=LCM of numerators HCF of denominators\text{LCM
of fractions} = \frac{\text{LCM of numerators}}{\text{HCF of
denominators}}LCM of fractions=HCF of denominatorsLCM of numerators
🔸 HCF of Fractions:
HCF of fractions=HCF of numeratorsLCM of denominators\text{HCF
of fractions} = \frac{\text{HCF of numerators}}{\text{LCM of
denominators}}HCF of fractions=LCM of denominatorsHCF of numerators
🔹 Applications in Word
Problems:
- Finding
the time when two or more repeating events will coincide → LCM
- Dividing
items/people into largest equal parts/groups → HCF
✅ Practice Questions (30 Total)
🔹 Easy Level (10
Questions)
- Find
the LCM of 4 and 5.
- Find
the HCF of 18 and 27.
- Find
the LCM and HCF of 8 and 12.
- Two
numbers are 16 and 24. What is their HCF?
- Find
the HCF of 30 and 45 using prime factorization.
- LCM
of two numbers is 60, and HCF is 5. If one number is 20, find the other.
- Find
the LCM of 6, 9, and 15.
- What
is the HCF of 0 and 12?
- Find
the LCM of 10 and 25.
- Find
the smallest number divisible by both 12 and 18.
🔹 Moderate Level (10
Questions)
- Find
the LCM and HCF of 72 and 120.
- The
HCF of two numbers is 8, and their LCM is 120. If one number is 24, find
the other.
- Find
the least number divisible by 5, 6, and 7.
- What
is the HCF of 48, 60, and 72?
- Find
the LCM of 3/4, 5/6, and 7/8.
- A
man has 168 apples and 252 oranges. He wants to pack them in boxes such
that each box has equal number of fruits and no fruit is left. What is the
maximum number of fruits in one box?
- Find
the HCF and LCM of 2² × 3³ × 5 and 2 × 3⁴ × 7.
- Two
numbers are in the ratio 5:7, and their HCF is 12. Find the numbers.
- The
product of two numbers is 1575 and their HCF is 15. Find their LCM.
- Find
the greatest number that exactly divides 135 and 225.
🔹 Difficult Level (10
Questions)
- Find
the HCF of 252 and 105 using the division method.
- The
HCF and LCM of two numbers are 16 and 240 respectively. If one number is
48, find the other.
- A
traffic light at three different road crossings changes after every 48
seconds, 72 seconds, and 108 seconds respectively. Find the time after
which all three signals change together.
- There
are three pieces of timber measuring 42 m, 49 m, and 63 m. What is the
greatest length that can be used to measure all the three pieces exactly?
- Three
runners start running together and complete a round in 36 sec, 48 sec, and
60 sec respectively. In how much time will they meet again at the starting
point?
- A
rectangular field of length 105 m and breadth 63 m is to be paved with
square tiles. What is the largest size of tile that can be used?
- Find
the least number which when divided by 35, 45, and 55 leaves remainder 5
in each case.
- Find
the largest number which divides 245 and 1029 leaving the same remainder
in each case.
- The
product of two numbers is 5040 and their HCF is 12. Find the numbers.
- Two
numbers are such that their difference is 36 and their HCF is 12. Find the
possible pairs of such numbers.
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