HCF and LCM: Key Concepts, Formulas, and Practice Questions

Find the HCF of 36 and 60.

Answer: 12
(Prime factors: 36 = $2^2 \times 3^2$22×32, 60 = $2^2 \times 3 \times 5$22×3×5; common factors = $2^2 \times 3$22×3).

What is the HCF of 45, 75, and 120?

Answer: 15
(Prime factors: 45 = $3^2 \times 5$32×5, 75 = $3 \times 5^2$3×52, 120 = $2^3 \times 3 \times 5$23×3×5; common factors = $3 \times 5$3×5).

Find the HCF of 54 and 90 using the prime factorization method.

Answer: 18
(Prime factors: 54 = $2 \times 3^3$2×33, 90 = $2 \times 3^2 \times 5$2×32×5; common factors = $2 \times 3^2$2×32).

The HCF of two numbers is 18, and their product is 540. If one of the numbers is 36, find the other number.

Answer: 30
(Use the formula: $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers, so $18 \times \text{LCM} = 36 \times 30$18×LCM=36×30; LCM = 30).

Find the HCF of 64 and 48 using the division method.

Answer: 16
(Divide 64 by 48 to get a remainder of 16. Then, divide 48 by 16, which gives remainder 0, so HCF = 16).

The HCF of two numbers is 8, and one of the numbers is 16. What is the other number?

Answer: 24
(Use the formula: $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers, so $8 \times \text{LCM} = 16 \times 24$8×LCM=16×24).

Find the HCF of 18, 24, and 30.

Answer: 6
(Prime factors: 18 = $2 \times 3^2$2×32, 24 = $2^3 \times 3$23×3, 30 = $2 \times 3 \times 5$2×3×5; common factors = $2 \times 3$2×3).

Find the greatest number that can divide 72 and 120 leaving a remainder of 4 in each case.

Answer: 4
(Subtract 4 from both numbers: 72-4 = 68, 120-4 = 116. HCF of 68 and 116 is 4).

What is the HCF of 20, 30, and 50?

Answer: 10
(Prime factors: 20 = $2^2 \times 5$22×5, 30 = $2 \times 3 \times 5$2×3×5, 50 = $2 \times 5^2$2×52; common factors = $2 \times 5$2×5).

Find the HCF of 35 and 70 using the division method.

Answer: 35
(Divide 70 by 35, and the remainder is 0, so HCF = 35).

LCM (Lowest Common Multiple)

11. Find the LCM of 24 and 36.
    • Answer: 72
      (Prime factors: 24 = $2^3 \times 3$23×3, 36 = $2^2 \times 3^2$22×32; take the highest powers: $2^3 \times 3^2$23×32).
12. What is the LCM of 15 and 20?
    • Answer: 60
      (Prime factors: 15 = $3 \times 5$3×5, 20 = $2^2 \times 5$22×5; take the highest powers: $2^2 \times 3 \times 5$22×3×5).
13. The LCM of two numbers is 120, and their HCF is 12. If one number is 24, find the other number.
    • Answer: 60
      (Use the formula: $\text{LCM} \times \text{HCF} = \text{Product of the numbers}$LCM×HCF=Product of the numbers, so $120 \times 12 = 24 \times \text{Other number}$120×12=24×Other number).
14. Find the LCM of 12, 18, and 24.
    • Answer: 72
      (Prime factors: 12 = $2^2 \times 3$22×3, 18 = $2 \times 3^2$2×32, 24 = $2^3 \times 3$23×3; take the highest powers: $2^3 \times 3^2$23×32).
15. What is the least common multiple of 45 and 60?
    • Answer: 180
      (Prime factors: 45 = $3^2 \times 5$32×5, 60 = $2^2 \times 3 \times 5$22×3×5; take the highest powers: $2^2 \times 3^2 \times 5$22×32×5).
16. Find the LCM of 50 and 75.
    • Answer: 150
      (Prime factors: 50 = $2 \times 5^2$2×52, 75 = $3 \times 5^2$3×52; take the highest powers: $2 \times 3 \times 5^2$2×3×52).
17. The LCM of two numbers is 180, and their HCF is 15. If one number is 45, find the other number.
    • Answer: 60
      (Use the formula: $\text{LCM} \times \text{HCF} = \text{Product of the numbers}$LCM×HCF=Product of the numbers).
18. Find the smallest number divisible by both 18 and 30.
    • Answer: 90
      (LCM of 18 and 30 = $2 \times 3^2 \times 5$2×32×5).
19. The LCM of two numbers is 72, and their HCF is 9. If one number is 18, find the other number.
    • Answer: 36
      (Use the formula: $\text{LCM} \times \text{HCF} = \text{Product of the numbers}$LCM×HCF=Product of the numbers).
20. Find the LCM of 8, 14, and 21.
    • Answer: 168
      (Prime factors: 8 = $2^3$23, 14 = $2 \times 7$2×7, 21 = $3 \times 7$3×7; take the highest powers: $2^3 \times 3 \times 7$23×3×7).

HCF and LCM Combined

Answer: 27 and 35
(Use the formula: $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers).

The product of two numbers is 360. If their HCF is 12, find their LCM.

Answer: 30
(Use the formula: $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers).

Find the LCM of 36 and 60, and also find their HCF.

Answer:

HCF = 12 (Common factors: $2^2 \times 3$22×3)

LCM = 180 (Highest powers: $2^3 \times 3^2 \times 5$23×32×5).

The HCF of two numbers is 14, and their LCM is 84. Find the numbers.

Answer: 28 and 42
(Use the formula: $\text{HCF} \times \text{LCM} = \text{Product of the numbers}$HCF×LCM=Product of the numbers).

Two bells ring together at 6:00 AM. One rings every 15 minutes, and the other rings every 20 minutes. When will they ring together again?

Answer: At 6:00 AM + 60 minutes = 7:00 AM
(LCM of 15 and 20 = 60).

The HCF of two numbers is 9, and their LCM is 315. Find the two numbers.

1. HCF (Highest Common Factor)

Definition:

• The HCF, also known as the GCD (Greatest Common Divisor), of two or more numbers is the largest number that divides all of them without leaving a remainder.

Properties of HCF:

• HCF is always less than or equal to the smallest of the numbers.
• HCF is used to simplify fractions.
• It helps in solving problems like dividing something into equal parts.

Methods to Find HCF:

• Prime Factorization Method:
  1. Express each number as a product of prime factors.
  2. The HCF is obtained by multiplying the common prime factors with the lowest powers.
• Division Method:
  1. Divide the larger number by the smaller number.
  2. Divide the divisor by the remainder.
  3. Continue until the remainder is 0. The divisor at this stage will be the HCF.

Example:
Find the HCF of 24 and 36.

• Prime factorization of 24 = $2^3 \times 3$23×3
• Prime factorization of 36 = $2^2 \times 3^2$22×32
• The common factors are $2^2$22 and $3$3.
• Hence, the HCF = $2^2 \times 3 = 12$22×3=12.

2. LCM (Lowest Common Multiple)

Definition:

• The LCM of two or more numbers is the smallest number that is divisible by all the given numbers.

Properties of LCM:

• LCM is always greater than or equal to the largest of the numbers.
• It is used in problems involving synchronization, such as finding the least time after which two events will coincide.

Methods to Find LCM:

• Prime Factorization Method:
  1. Express each number as a product of prime factors.
  2. The LCM is obtained by multiplying the highest powers of all prime factors that appear in the factorization of each number.
• Division Method:
  1. Divide the numbers by their common prime factors.
  2. Continue dividing until all the numbers are reduced to 1.
  3. Multiply the divisors at each step to get the LCM.

Example:
Find the LCM of 24 and 36.

• Prime factorization of 24 = $2^3 \times 3$23×3
• Prime factorization of 36 = $2^2 \times 3^2$22×32
• The LCM is obtained by taking the highest powers of all prime factors: $2^3 \times 3^2$23×32.
• Hence, the LCM = $2^3 \times 3^2 = 72$23×32=72.

Relationship Between HCF and LCM

There is a useful relationship between HCF and LCM for two numbers:$$\text{HCF} \times \text{LCM} = \text{Product of the two numbers}$$HCF×LCM=Product of the two numbers

Example:
For the numbers 24 and 36:

• We know from earlier that HCF = 12 and LCM = 72.
• The product of the two numbers is: $24 \times 36 = 864$24×36=864.
• Check: $12 \times 72 = 864$12×72=864, which is correct.

3. Applications of HCF and LCM

1. HCF in Fraction Reduction:
   • To reduce a fraction to its simplest form, divide both the numerator and the denominator by their HCF.
   • Example: $\frac{36}{60}$6036​. The HCF of 36 and 60 is 12, so $\frac{36}{60} = \frac{36 \div 12}{60 \div 12} = \frac{3}{5}$6036​=60÷1236÷12​=53​.
2. LCM in Finding Common Time:
   • LCM is used to determine when two or more events will coincide.
   • Example: Two buses leave at 8 AM. One bus departs every 24 minutes, and the other departs every 36 minutes. To find when they will leave together again, calculate the LCM of 24 and 36. LCM = 72, so the buses will leave together again after 72 minutes (i.e., at 9:12 AM).

4. Quick Tricks for HCF and LCM

Example: HCF of 5 and 6 = 1, LCM of 5 and 6 = 30.

For HCF: If you are given more than two numbers, find the HCF of the first two numbers first, then use that result to find the HCF with the next number, and so on.

For LCM: If you are given more than two numbers, find the LCM of the first two numbers first, then use that result to find the LCM with the next number, and so on.

HCF and LCM of Consecutive Numbers: For any two consecutive numbers, their HCF will always be 1 and their LCM will be the product of the numbers.