Answer :
To find the highest common factor (HCF) of the numbers 72, 90, and 126, we can follow these steps:
1. Find the prime factors of each number:
- 72 can be factored as:
[tex]\(72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\)[/tex]
- 90 can be factored as:
[tex]\(90 = 2 \times 3 \times 3 \times 5 = 2^1 \times 3^2 \times 5^1\)[/tex]
- 126 can be factored as:
[tex]\(126 = 2 \times 3 \times 3 \times 7 = 2^1 \times 3^2 \times 7^1\)[/tex]
2. Identify the common prime factors:
- The numbers all have the prime factors 2 and 3 in common.
3. Choose the smallest powers of these common factors:
- For the factor 2, the smallest power is [tex]\(2^1\)[/tex] (since 72 has [tex]\(2^3\)[/tex], and both 90 and 126 have [tex]\(2^1\)[/tex]).
- For the factor 3, the smallest power is [tex]\(3^2\)[/tex] (all numbers have [tex]\(3^2\)[/tex]).
4. Multiply these factors together to find the HCF:
- [tex]\(HCF = 2^1 \times 3^2 = 2 \times 9 = 18\)[/tex]
So, the highest common factor of 72, 90, and 126 is 18.
1. Find the prime factors of each number:
- 72 can be factored as:
[tex]\(72 = 2 \times 2 \times 2 \times 3 \times 3 = 2^3 \times 3^2\)[/tex]
- 90 can be factored as:
[tex]\(90 = 2 \times 3 \times 3 \times 5 = 2^1 \times 3^2 \times 5^1\)[/tex]
- 126 can be factored as:
[tex]\(126 = 2 \times 3 \times 3 \times 7 = 2^1 \times 3^2 \times 7^1\)[/tex]
2. Identify the common prime factors:
- The numbers all have the prime factors 2 and 3 in common.
3. Choose the smallest powers of these common factors:
- For the factor 2, the smallest power is [tex]\(2^1\)[/tex] (since 72 has [tex]\(2^3\)[/tex], and both 90 and 126 have [tex]\(2^1\)[/tex]).
- For the factor 3, the smallest power is [tex]\(3^2\)[/tex] (all numbers have [tex]\(3^2\)[/tex]).
4. Multiply these factors together to find the HCF:
- [tex]\(HCF = 2^1 \times 3^2 = 2 \times 9 = 18\)[/tex]
So, the highest common factor of 72, 90, and 126 is 18.
