$$ A=2^{2} imes 3^{5} imes 5 \quad B=2^{4} imes 3^{2} imes 5^{3} $$ Write down the highest common factor (HCF) of $$ A $$ and $$ B $$.

Question

$$ A=2^{2} 	imes 3^{5} 	imes 5 \quad B=2^{4} 	imes 3^{2} 	imes 5^{3} $$ Write down the highest common factor (HCF) of $$ A $$ and $$ B $$.

UpStudy AI · Accepted Answer

To find the highest common factor (HCF) of $$ A $$ and $$ B $$, we need to identify the minimum power of each prime factor present in both expressions.

Given:
- $$ A = 2^{2} \times 3^{5} \times 5^{1} $$
- $$ B = 2^{4} \times 3^{2} \times 5^{3} $$

Now, let's identify the minimum powers of each prime factor:

1. For the prime factor $$ 2 $$:
   - In $$ A $$, the power is $$ 2 $$.
   - In $$ B $$, the power is $$ 4 $$.
   - Minimum power: $$ \min(2, 4) = 2 $$.

2. For the prime factor $$ 3 $$:
   - In $$ A $$, the power is $$ 5 $$.
   - In $$ B $$, the power is $$ 2 $$.
   - Minimum power: $$ \min(5, 2) = 2 $$.

3. For the prime factor $$ 5 $$:
   - In $$ A $$, the power is $$ 1 $$.
   - In $$ B $$, the power is $$ 3 $$.
   - Minimum power: $$ \min(1, 3) = 1 $$.

Now, we can write the HCF using the minimum powers of the prime factors:

$$
\text{HCF}(A, B) = 2^{2} \times 3^{2} \times 5^{1}
$$

Let's calculate this expression:
Calculate the value by following steps:
- step0: Calculate:
  $$2^{2}\times 3^{2}\times 5^{1}$$
- step1: Evaluate the power:
  $$2^{2}\times 3^{2}\times 5$$
- step2: Multiply the numbers:
  $$6^{2}\times 5$$
- step3: Evaluate the power:
  $$36\times 5$$
- step4: Multiply:
  $$180$$
The highest common factor (HCF) of $$ A $$ and $$ B $$ is $$ 2^{2} \times 3^{2} \times 5^{1} = 180 $$.
The highest common factor (HCF) of $$ A $$ and $$ B $$ is 180.

Write down the highest common factor (HCF) of and .

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