`(561)/(748)` when reduced to lowest terms is-

Options

A. `13/14`
B. `3/4`
C. `11/14`
D. `23/24`

Answer

To reduce the fraction \(\frac{561}{748}\) to its lowest terms, we will follow these steps:

### Step 1: Find the Prime Factorization of the Numerator (561)
1. Start by dividing 561 by the smallest prime number, which is 3.
- \(561 \div 3 = 187\)
- So, \(561 = 3 \times 187\).

2. Next, we need to check if 187 can be factored further.
- Check divisibility by 11 (next prime number):
- \(187 \div 11 \approx 17\) (not divisible).
- Check divisibility by 17:
- \(187 \div 17 = 11\) (exactly divisible).
- Thus, \(187 = 11 \times 17\).

3. Therefore, the complete factorization of 561 is:
- \(561 = 3 \times 11 \times 17\).

### Step 2: Find the Prime Factorization of the Denominator (748)
1. Start by dividing 748 by the smallest prime number, which is 2.
- \(748 \div 2 = 374\)
- So, \(748 = 2 \times 374\).

2. Continue factoring 374:
- \(374 \div 2 = 187\)
- So, \(374 = 2 \times 187\).
- Therefore, \(748 = 2^2 \times 187\).

3. We already factored 187 in the previous step:
- \(187 = 11 \times 17\).
- Thus, \(748 = 2^2 \times 11 \times 17\).

### Step 3: Write the Fraction with Prime Factors
Now we can write the fraction \(\frac{561}{748}\) using its prime factors:
\[ \frac{561}{748} = \frac{3 \times 11 \times 17}{2^2 \times 11 \times 17} \]

### Step 4: Cancel Common Factors
We can cancel the common factors of \(11\) and \(17\) from the numerator and denominator:
\[ \frac{3 \times \cancel{11} \times \cancel{17}}{2^2 \times \cancel{11} \times \cancel{17}} = \frac{3}{2^2} = \frac{3}{4} \]

### Step 5: Final Answer
Thus, the fraction \(\frac{561}{748}\) reduced to its lowest terms is:
\[ \frac{3}{4} \]

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