This topic would deal with finding the highest common factor using the prime factorization method. Hence, we will find the prime factors of the number using short division method and then the highest common factor would be determined.
Steps to find out the highest common factor using the prime factorization method:
Step I: First find out the prime factors of the numbers given
Step II: Then enlist the common prime factors of the numbers
Step III: Then find out the lowest power of those common prime factors
Step IV: Then find out the product of the lowest power of those prime numbers
Here are few examples illustrated to find out the highest common factor by using prime factorization method:
1. Find out the highest common factor of 15 and 20 by prime factorization method:
Solution:
The prime factors of 15 are: 5 × 3
The prime factors of 20 are: 5 × 5 × 2 × 2 = 5^{2 }× 2^{2 }
The common prime factor of 15 and 20 is 5
The lowest power of 5 is 5
Hence, H.C.F of 15 and 20 = 5
2. Find out the highest common factor of 24 and 12 by prime factorization method:
Solution:
The prime factors of 24 are: 2 × 2 × 2 × 3 = 2^{3} × 3
The prime factors of 12 are: 2 × 2 × 3 = 2^{2} × 3
The common prime factors of 24 and 12 are = 2 and 3
The lowest power of 2 = 2^{2}
The lowest power of 3 is 3
The H. C. F of 24 and 12 = 2^{2} × 3 = 2 × 2 × 3 = 12
3. Find out the highest common factor of 18 and 30 by prime factorization method:
Solution:
The prime factors of 18 are: 2 × 3 × 3 = 2 × 3^{2}
The prime factors of 30 are: 2 × 3 × 5
The common prime factors of 18 and 30 are: 2 and 3
The lowest power of 2 is 2
The lowest power of 3 is 3
The H. C. F of 18 and 30 = 2 × 3 = 6
4. Find out the highest common factor of 56 and 35 by prime factorization method:
Solution:
The prime factors of 56 are: 2 × 2 × 2 × 7 = 2^{3} × 7
The prime factors of 35 are: 5 × 7
The common prime factors of 56 and 35 is 7
The lowest power of 7 is 7
The H. C. F of 56 and 35 = 7
4. Find out the highest common factor of 105 and 110 by prime factorization method:
Solution:
The prime factors of 105 = 5 × 3 × 7
The prime factors of 110 = 2 × 5 × 11
The common prime factors of 105 and 110 is 5
The lowest power of 5 is 5
Therefore, the H.C.F of 105 and 110 is 5
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