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Find the highest common factor (HCF) of 84 and 105.

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To find the highest common factor (HCF) of 84 and 105, we follow these steps:

1. Prime Factorization:
- First, we find the prime factorization of each number.

- For 84:
- 84 is divisible by 2: [tex]\( 84 \div 2 = 42 \)[/tex]
- 42 is divisible by 2: [tex]\( 42 \div 2 = 21 \)[/tex]
- 21 is divisible by 3: [tex]\( 21 \div 3 = 7 \)[/tex]
- 7 is a prime number.

So, the prime factorization of 84 is:
[tex]\[ 84 = 2^2 \times 3 \times 7 \][/tex]

- For 105:
- 105 is divisible by 3: [tex]\( 105 \div 3 = 35 \)[/tex]
- 35 is divisible by 5: [tex]\( 35 \div 5 = 7 \)[/tex]
- 7 is a prime number.

So, the prime factorization of 105 is:
[tex]\[ 105 = 3 \times 5 \times 7 \][/tex]

2. Identify Common Factors:
- The common prime factors between 84 and 105 are 3 and 7.

3. Multiply the Common Factors:
- To find the HCF, we multiply all the common prime factors together:
[tex]\[ HCF = 3 \times 7 = 21 \][/tex]

Therefore, the highest common factor (HCF) of 84 and 105 is 21.
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