Sure! I'll walk you through finding the Highest Common Factor (H.C.F.), also known as the Greatest Common Divisor (G.C.D.), of each set of numbers using the factorization method.
### (a) 18, 24
1. Prime factorize each number:
- 18 = 2 × 3²
- 24 = 2³ × 3
2. Identify the common factors:
- Common prime factors: 2 and 3
- Lowest power of common prime factors: 2¹ and 3¹
3. Multiply the lowest powers of the common factors:
- H.C.F. = 2 × 3 = 6
### (b) 40, 50
1. Prime factorize each number:
- 40 = 2³ × 5
- 50 = 2 × 5²
2. Identify the common factors:
- Common prime factors: 2 and 5
- Lowest power of common prime factors: 2¹ and 5¹
3. Multiply the lowest powers of the common factors:
- H.C.F. = 2 × 5 = 10
### (c) 54, 72
1. Prime factorize each number:
- 54 = 2 × 3³
- 72 = 2³ × 3²
2. Identify the common factors:
- Common prime factors: 2 and 3
- Lowest power of common prime factors: 2¹ and 3²
3. Multiply the lowest powers of the common factors:
- H.C.F. = 2 × 3² = 2 × 9 = 18
### (d) 24, 36, 42
1. Prime factorize each number:
- 24 = 2³ × 3
- 36 = 2² × 3²
- 42 = 2 × 3 × 7
2. Identify the common factors:
- Common prime factors: 2 and 3
- Lowest power of common prime factors: 2¹ and 3¹
3. Multiply the lowest powers of the common factors:
- H.C.F. = 2 × 3 = 6
### (e) 35, 56, 70
1. Prime factorize each number:
- 35 = 5 × 7
- 56 = 2³ × 7
- 70 = 2 × 5 × 7
2. Identify the common factors:
- Common prime factor: 7
- Lowest power of common prime factor: 7¹
3. Multiply the lowest power of the common factor:
- H.C.F. = 7
### (f) 30, 75, 90
1. Prime factorize each number:
- 30 = 2 × 3 × 5
- 75 = 3 × 5²
- 90 = 2 × 3² × 5
2. Identify the common factors:
- Common prime factors: 3 and 5
- Lowest power of common prime factors: 3¹ and 5¹
3. Multiply the lowest powers of the common factors:
- H.C.F. = 3 × 5 = 15
### (g) 32, 64, 80
1. Prime factorize each number:
- 32 = 2⁵
- 64 = 2⁶
- 80 = 2⁴ × 5
2. Identify the common factor:
- Common prime factor: 2
- Lowest power of common prime factor: 2⁴
3. The common prime factor with the lowest power is:
- H.C.F. = 2⁴ = 16
### (h) 96, 144, 216
1. Prime factorize each number:
- 96 = 2⁵ × 3
- 144 = 2⁴ × 3²
- 216 = 2³ × 3³
2. Identify the common factors:
- Common prime factors: 2 and 3
- Lowest power of common prime factors: 2³ and 3¹
3. Multiply the lowest powers of the common factors:
- H.C.F. = 2³ × 3 = 8 × 3 = 24
Thus, the H.C.F. calculations for each set are:
- (a) 18, 24 => 6
- (b) 40, 50 => 10
- (c) 54, 72 => 18
- (d) 24, 36, 42 => 6
- (e) 35, 56, 70 => 7
- (f) 30, 75, 90 => 15
- (g) 32, 64, 80 => 16
- (h) 96, 144, 216 => 24
