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Answer :
To find the highest common factor (HCF) of the numbers 72, 90, and 126, follow these steps:
1. Prime Factorization:
- First, break down each number into its prime factors.
- 72:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
- 90:
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- So, the prime factorization of 90 is [tex]\(2^1 \times 3^2 \times 5^1\)[/tex].
- 126:
- 126 ÷ 2 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- So, the prime factorization of 126 is [tex]\(2^1 \times 3^2 \times 7^1\)[/tex].
2. Identify Common Prime Factors:
- Now, look for prime factors that are common in the factorizations of all three numbers.
- The common prime factors of 72, 90, and 126 are:
- The factor [tex]\(2\)[/tex] appears in [tex]\(2^1\)[/tex].
- The factor [tex]\(3\)[/tex] appears in [tex]\(3^2\)[/tex].
3. Calculate the HCF:
- The HCF is obtained by multiplying the smallest power of all common prime factors.
- In this case:
- For [tex]\(2\)[/tex], the smallest power is [tex]\(2^1\)[/tex].
- For [tex]\(3\)[/tex], the smallest power is [tex]\(3^2\)[/tex].
- Therefore, the HCF is [tex]\(2^1 \times 3^2 = 2 \times 9 = 18\)[/tex].
So, the highest common factor of 72, 90, and 126 is 18.
1. Prime Factorization:
- First, break down each number into its prime factors.
- 72:
- 72 ÷ 2 = 36
- 36 ÷ 2 = 18
- 18 ÷ 2 = 9
- 9 ÷ 3 = 3
- 3 ÷ 3 = 1
- So, the prime factorization of 72 is [tex]\(2^3 \times 3^2\)[/tex].
- 90:
- 90 ÷ 2 = 45
- 45 ÷ 3 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- So, the prime factorization of 90 is [tex]\(2^1 \times 3^2 \times 5^1\)[/tex].
- 126:
- 126 ÷ 2 = 63
- 63 ÷ 3 = 21
- 21 ÷ 3 = 7
- 7 ÷ 7 = 1
- So, the prime factorization of 126 is [tex]\(2^1 \times 3^2 \times 7^1\)[/tex].
2. Identify Common Prime Factors:
- Now, look for prime factors that are common in the factorizations of all three numbers.
- The common prime factors of 72, 90, and 126 are:
- The factor [tex]\(2\)[/tex] appears in [tex]\(2^1\)[/tex].
- The factor [tex]\(3\)[/tex] appears in [tex]\(3^2\)[/tex].
3. Calculate the HCF:
- The HCF is obtained by multiplying the smallest power of all common prime factors.
- In this case:
- For [tex]\(2\)[/tex], the smallest power is [tex]\(2^1\)[/tex].
- For [tex]\(3\)[/tex], the smallest power is [tex]\(3^2\)[/tex].
- Therefore, the HCF is [tex]\(2^1 \times 3^2 = 2 \times 9 = 18\)[/tex].
So, the highest common factor of 72, 90, and 126 is 18.
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