This topic would deal with finding the highest common factor using the prime factorization method. With the help of this method we will first find the prime factors of the numbers and then sort out the common factor. Next step is to find the lowest power of those common factors and find out their product. That product would be the highest common factor.
Here are few examples to illustrate on finding the highest common factor using the prime factorization method:
1. Find out the highest common
factor of 256 and 224 by prime factorization method:
Solution:
The prime factors of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{8}
The prime factors of 224 = 2 × 2 × 2 × 2 × 2 × 7 = 2^{5} × 7
The common prime factor of 256 and 224 is 2
The lowest power of 2 is 2^{5}
Therefore, the H.C.F of 256 and 224 is 2^{5} = 2 × 2 × 2 × 2 × 2 = 32
2. Find out the highest common factor of 24 and 25 by prime factorization method:
Solution:
The prime factors of 24 = 2 × 2 × 2 × 3 = 2^{3} × 3
The prime factors of 25 = 5 × 5 = 5^{2}
The common prime factor of 24 and 25 is 1
Therefore, the H.C.F of 24 and 25 is 1
3. Find out the highest common factor of 105 and 124 by prime factorization method:
Solution:
The prime factors of 105 = 5 × 3 × 7
The prime factors of 124 = 2 × 2 × 31 = 2^{2} × 31
The common prime factor of 105 and 124 is 1
Therefore, the H.C.F of 105 and 124 is 1
4. Find out the highest common factor of 155 and 55 by prime factorization method:
Solution:
The prime factors of 155 = 5 × 31
The prime factors of 55 = 5 × 11
The common prime factor of 155 and 55 is 5
The lowest power of 5 is 5
Therefore, the H.C.F of 155 and 55 is 5
5. Find out the highest common factor of 48 and 60 by prime factorization method:
Solution:
The prime factors of 48 = 2 × 2 × 2 × 2 × 3 = 2^{4} × 3
The prime factors of 60 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5
The common prime factors of 48 and 60 are 2 and 3
The lowest power of 2 is 2^{2}
The lowest power of 3 is 3
Therefore, the H.C.F of 48 and 60 is 2^{2} × 3 = 2 × 2 × 3 = 12
6. Find out the highest common factor of 12 and 20 by prime factorization method:
Solution:
The prime factors of 12 = 2 × 2 × 3 = 2^{2} × 3
The prime factors of 20 = 2 × 2 × 5 = 2^{2 }× 5
The common prime factor of 48 and 60 is 2
The lowest power of 2 is 2^{2}
Therefore, the H.C.F of 12 and 20 is 2^{2 } = 2 × 2 = 4
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