This topic would discuss about how to find the least common multiple by using prime factorization method.
Steps to find out the least common multiple by using prime factorization method:
Step I: First find all the prime factors of the given number
Step II: Then write all the prime factors in exponential form
Step III: Then find the product of highest power of all the prime factors.
Here are few examples to illustrate least common multiple by using prime factorization method:
1. Find the least common factor of 256 and 224 by using prime factorization method.
The factors of 256 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 = 2^{8}
The factors of 224 = 2 × 2 × 2 × 2 × 2 × 7 = 2^{5} × 7
The highest power of 2 is 2^{8}
The highest power of 7 is 7
The least common multiple of 256 and 224 = 2^{8} × 5 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 7 = 1792
2. Find the least common factor of 105 and 110 by using prime factorization method
The factors of 105 = 5 × 3 × 7
The factors of 110 = 2 × 5 × 11
The highest power of 2 is 2
The highest power of 3 is 3
The highest power of 5 is 5
The highest power of 7 is 7
The highest power of 11 is 11
The least common multiple of 105 and 110 = 2 × 3 × 5 × 7 ×11 = 2310
Note:
Here we can see that there are no common factors hence all the factors of both the number are required to be multiplied to find the L.C.M
3. Find the least common factor of 48 and 60 by using prime factorization method
The factors of 48 = 2 × 2 × 2 × 2 × 3 = 2^{4} × 3
The factors of 60 = 2 × 2 × 3 × 5 = 2^{2} × 3 × 5
The highest power of 2 is 2^{4}
The highest power of 3 is 3
The highest power of 5 is 5
The least common multiple of 48 and 60 = 2^{4} × 5 × 3 = 2 × 2 × 2 × 2 × 3 × 5 = 240
4. Find the least common factor of 21 and 42 by using prime factorization method.
The factors of 21 = 3 × 7
The factors of 42 = 2 × 3 × 7
The highest power of 2 is 2
The highest power of 3 is 3
The highest power of 7 is 7
The least common multiple of 21 and 42 = 2 × 3 × 7 = 42
5. Find the least common factor of 44 and 88 by using prime factorization method.
The factors of 44 are: 2 × 2 × 11 = 2^{2} × 11
The factors of 88 are: 2 × 2 × 2 × 11 = 2^{3} × 11
The highest power of 2 is 2^{3} × 11
The highest power of 11 is 11
The least common multiple of 44 and 88 is 2 × 2 × 2 × 11 = 2^{3} × 11 = 88.
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