To find the highest common factor (HCF) of the numbers \(A\) and \(B\), where:
[tex]\[
A = 2^2 \times 3^1 \times 5^1
\][/tex]
[tex]\[
B = 2^3 \times 3^2 \times 5^1
\][/tex]
we need to follow these steps:
1. Identify the prime factors of each number with their respective exponents:
- The prime factorization of \(A\) is \(2^2 \times 3^1 \times 5^1\).
- The prime factorization of \(B\) is \(2^3 \times 3^2 \times 5^1\).
2. List the common prime factors and their smallest exponents from each factorization:
- For \(2\): The exponents are 2 (from \(A\)) and 3 (from \(B\)). The minimum exponent is 2.
- For \(3\): The exponents are 1 (from \(A\)) and 2 (from \(B\)). The minimum exponent is 1.
- For \(5\): The exponents are both 1 in \(A\) and \(B\). The minimum exponent is 1.
3. Construct the HCF by multiplying these primes raised to their smallest exponents:
- The smallest exponent for 2 is 2, hence \(2^2\).
- The smallest exponent for 3 is 1, hence \(3^1\).
- The smallest exponent for 5 is 1, hence \(5^1\).
4. Calculate the HCF:
[tex]\[
HCF = 2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 12 \times 5 = 60
\][/tex]
So, the highest common factor (HCF) of [tex]\(A\)[/tex] and [tex]\(B\)[/tex] is [tex]\(60\)[/tex].
