Question
least common denominator of 1/16, 3/56, and 5/112
Answers
GPT-4o mini
To find the least common denominator (LCD) of the fractions \( \frac{1}{16} \), \( \frac{3}{56} \), and \( \frac{5}{112} \), we first need to determine the denominators: 16, 56, and 112.
Next, we'll find the prime factorization of each denominator:
- \( 16 = 2^4 \)
- \( 56 = 2^3 \times 7 \)
- \( 112 = 2^4 \times 7 \)
Now, we take the highest power of each prime factor present in any of the factorizations:
- For the prime \( 2 \), the highest power is \( 2^4 \) (from 16 and 112).
- For the prime \( 7 \), the highest power is \( 7^1 \) (from 56 and 112).
Now, we multiply these together to get the least common denominator:
\[
LCD = 2^4 \times 7^1 = 16 \times 7 = 112
\]
Thus, the least common denominator of \( \frac{1}{16} \), \( \frac{3}{56} \), and \( \frac{5}{112} \) is \( \boxed{112} \).
Next, we'll find the prime factorization of each denominator:
- \( 16 = 2^4 \)
- \( 56 = 2^3 \times 7 \)
- \( 112 = 2^4 \times 7 \)
Now, we take the highest power of each prime factor present in any of the factorizations:
- For the prime \( 2 \), the highest power is \( 2^4 \) (from 16 and 112).
- For the prime \( 7 \), the highest power is \( 7^1 \) (from 56 and 112).
Now, we multiply these together to get the least common denominator:
\[
LCD = 2^4 \times 7^1 = 16 \times 7 = 112
\]
Thus, the least common denominator of \( \frac{1}{16} \), \( \frac{3}{56} \), and \( \frac{5}{112} \) is \( \boxed{112} \).