To find the ratio of the HCF (Highest Common Factor) and LCM (Lowest Common Multiple) of the numbers 52 and 130, we will follow these steps:
### Step 1: Find the prime factorization of both numbers.
- **For 52**:
- 52 can be divided by 2: \( 52 \div 2 = 26 \)
- 26 can be divided by 2: \( 26 \div 2 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 52 is \( 2^2 \times 13^1 \).
- **For 130**:
- 130 can be divided by 2: \( 130 \div 2 = 65 \)
- 65 can be divided by 5: \( 65 \div 5 = 13 \)
- 13 is a prime number.
- Therefore, the prime factorization of 130 is \( 2^1 \times 5^1 \times 13^1 \).
### Step 2: Find the HCF (Highest Common Factor).
- The HCF is found by taking the lowest power of all common prime factors.
- Common prime factors of 52 and 130 are:
- \( 2 \) (minimum power is \( 1 \))
- \( 13 \) (minimum power is \( 1 \))
Thus, the HCF is:
\[
HCF = 2^1 \times 13^1 = 2 \times 13 = 26
\]
### Step 3: Find the LCM (Lowest Common Multiple).
- The LCM is found by taking the highest power of all prime factors present in either number.
- Prime factors are:
- \( 2 \) (maximum power is \( 2 \))
- \( 5 \) (maximum power is \( 1 \))
- \( 13 \) (maximum power is \( 1 \))
Thus, the LCM is:
\[
LCM = 2^2 \times 5^1 \times 13^1 = 4 \times 5 \times 13
\]
Calculating this:
\[
4 \times 5 = 20
\]
\[
20 \times 13 = 260
\]
So, the LCM is \( 260 \).
### Step 4: Find the ratio of HCF to LCM.
Now we can find the ratio:
\[
\text{Ratio} = \frac{HCF}{LCM} = \frac{26}{260}
\]
Simplifying this:
\[
\frac{26}{260} = \frac{1}{10}
\]
### Final Answer:
The ratio of the HCF and LCM of 52 and 130 is \( \frac{1}{10} \).
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