Answer: (a) 1 , (b) 2 , (c) 4
Explanation:
To find the highest common factor (HCF) of the given numbers, you can use both the long division method and the prime factorization method. Let's find the HCF for each set of numbers:
(a) 81 and 54:
Long Division Method:
- Divide 81 by 54:
81 ÷ 54 = 1 with a remainder of 27
- Now, divide the divisor (54) by the remainder (27):
54 ÷ 27 = 2
- Continue this process:
27 ÷ 2 = 13 with a remainder of 1
- Finally, divide 2 by 1:
2 ÷ 1 = 2
The HCF is 1.
Prime Factorization Method:
- Find the prime factors of both numbers:
81 = 3^4
54 = 2 * 3^3
- Identify the common prime factors and take the lowest power:
Common factor = 3^3
So, HCF = 3^3 = 27
(b) 96 and 44:
Long Division Method:
- Divide 96 by 44:
96 ÷ 44 = 2 with a remainder of 8
- Now, divide the divisor (44) by the remainder (8):
44 ÷ 8 = 5 with a remainder of 4
- Continue this process:
8 ÷ 4 = 2
- Finally, divide 4 by 2:
4 ÷ 2 = 2
The HCF is 2.
Prime Factorization Method:
- Find the prime factors of both numbers:
96 = 2^5 * 3
44 = 2^2 * 11
- Identify the common prime factors and take the lowest power:
Common factor = 2^2
So, HCF = 2^2 = 4
(c) 20, 28, and 36:
Prime Factorization Method:
- Find the prime factors of each number:
20 = 2^2 * 5
28 = 2^2 * 7
36 = 2^2 * 3^2
- Identify the common prime factors and take the lowest power:
Common factor = 2^2
So, HCF = 2^2 = 4
Therefore, the HCF for each set of numbers is:
(a) 1
(b) 2
(c) 4