To find the Highest Common Factor (H.C.F.) of the given numbers using the prime factorization method, we will follow these steps for each pair of numbers.
### (i) H.C.F. of 28 and 46
**Step 1: Prime Factorization of 28**
- 28 can be factorized as:
- 28 = 2 × 14
- 14 = 2 × 7
- Therefore, the prime factorization of 28 is:
- 28 = 2² × 7
**Step 2: Prime Factorization of 46**
- 46 can be factorized as:
- 46 = 2 × 23
- Therefore, the prime factorization of 46 is:
- 46 = 2 × 23
**Step 3: Identify Common Factors**
- The common prime factor between 28 and 46 is:
- 2
**Step 4: H.C.F.**
- The H.C.F. of 28 and 46 is:
- H.C.F. = 2
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### (ii) H.C.F. of 25 and 65
**Step 1: Prime Factorization of 25**
- 25 can be factorized as:
- 25 = 5 × 5
- Therefore, the prime factorization of 25 is:
- 25 = 5²
**Step 2: Prime Factorization of 65**
- 65 can be factorized as:
- 65 = 5 × 13
- Therefore, the prime factorization of 65 is:
- 65 = 5 × 13
**Step 3: Identify Common Factors**
- The common prime factor between 25 and 65 is:
- 5
**Step 4: H.C.F.**
- The H.C.F. of 25 and 65 is:
- H.C.F. = 5
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### (iii) H.C.F. of 54, 72, and 90
**Step 1: Prime Factorization of 54**
- 54 can be factorized as:
- 54 = 2 × 27
- 27 = 3 × 9
- 9 = 3 × 3
- Therefore, the prime factorization of 54 is:
- 54 = 2 × 3³
**Step 2: Prime Factorization of 72**
- 72 can be factorized as:
- 72 = 8 × 9
- 8 = 2³
- 9 = 3²
- Therefore, the prime factorization of 72 is:
- 72 = 2³ × 3²
**Step 3: Prime Factorization of 90**
- 90 can be factorized as:
- 90 = 9 × 10
- 9 = 3²
- 10 = 2 × 5
- Therefore, the prime factorization of 90 is:
- 90 = 2 × 3² × 5
**Step 4: Identify Common Factors**
- The common prime factors are:
- 2 (minimum power is 1)
- 3 (minimum power is 2)
**Step 5: H.C.F.**
- The H.C.F. of 54, 72, and 90 is:
- H.C.F. = 2¹ × 3² = 2 × 9 = 18
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### (iv) H.C.F. of 105, 140, and 175
**Step 1: Prime Factorization of 105**
- 105 can be factorized as:
- 105 = 3 × 35
- 35 = 5 × 7
- Therefore, the prime factorization of 105 is:
- 105 = 3 × 5 × 7
**Step 2: Prime Factorization of 140**
- 140 can be factorized as:
- 140 = 14 × 10
- 14 = 2 × 7
- 10 = 2 × 5
- Therefore, the prime factorization of 140 is:
- 140 = 2² × 5 × 7
**Step 3: Prime Factorization of 175**
- 175 can be factorized as:
- 175 = 25 × 7
- 25 = 5 × 5
- Therefore, the prime factorization of 175 is:
- 175 = 5² × 7
**Step 4: Identify Common Factors**
- The common prime factors are:
- 5 (minimum power is 1)
- 7 (minimum power is 1)
**Step 5: H.C.F.**
- The H.C.F. of 105, 140, and 175 is:
- H.C.F. = 5¹ × 7¹ = 5 × 7 = 35
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