To find the highest common factor (HCF) of the numbers [tex]\( A \)[/tex] and [tex]\( B \)[/tex], we follow these steps:
1. Express the numbers as a product of their prime factors:
[tex]\[
A = 2^2 \times 3^3 \times 5^2
\][/tex]
[tex]\[
B = 2 \times 3 \times 5^2 \times 7
\][/tex]
2. Identify the common prime factors:
- The prime factors of [tex]\( A \)[/tex] are 2, 3, and 5.
- The prime factors of [tex]\( B \)[/tex] are 2, 3, 5, and 7.
- The common factors between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] are 2, 3, and 5.
3. Determine the lowest power of each common prime factor that appears in both factorizations:
- For the prime factor [tex]\( 2 \)[/tex]:
- In [tex]\( A \)[/tex], the exponent is [tex]\( 2 \)[/tex].
- In [tex]\( B \)[/tex], the exponent is [tex]\( 1 \)[/tex].
- The lowest power is [tex]\( \min(2, 1) = 1 \)[/tex].
- For the prime factor [tex]\( 3 \)[/tex]:
- In [tex]\( A \)[/tex], the exponent is [tex]\( 3 \)[/tex].
- In [tex]\( B \)[/tex], the exponent is [tex]\( 1 \)[/tex].
- The lowest power is [tex]\( \min(3, 1) = 1 \)[/tex].
- For the prime factor [tex]\( 5 \)[/tex]:
- In [tex]\( A \)[/tex], the exponent is [tex]\( 2 \)[/tex].
- In [tex]\( B \)[/tex], the exponent is [tex]\( 2 \)[/tex].
- The lowest power is [tex]\( \min(2, 2) = 2 \)[/tex].
4. Form the HCF by multiplying these lowest powers of the common prime factors:
[tex]\[
\text{HCF} = 2^1 \times 3^1 \times 5^2
\][/tex]
5. Calculate the HCF:
[tex]\[
2^1 = 2
\][/tex]
[tex]\[
3^1 = 3
\][/tex]
[tex]\[
5^2 = 25
\][/tex]
[tex]\[
\text{HCF} = 2 \times 3 \times 25 = 2 \times 3 = 6
\][/tex]
[tex]\[
6 \times 25 = 150
\][/tex]
The highest common factor (HCF) of [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[
\boxed{150}
\][/tex]
