To find the Least Common Denominator (LCD) of the fractions $\frac{4}{7}, \frac{4}{14}, \frac{5}{21}$, we need to determine the Least Common Multiple (LCM) of the denominators: 7, 14, and 21.

Step-by-step process:

1. Prime factorization of the denominators:

   • 7: $7 = 7$
   • 14: $14 = 2 \times 7$
   • 21: $21 = 3 \times 7$

2. Find the LCM:

   • Take the highest power of each prime factor that appears in any of the factorizations.
     • For 2, the highest power is $2^1$ (from 14).
     • For 3, the highest power is $3^1$ (from 21).
     • For 7, the highest power is $7^1$ (common in all factorizations).
   • Multiply these together: $2^1 \times 3^1 \times 7^1 = 42$.

Thus, the Least Common Denominator (LCD) is 42.

Would you like to see more details or have any other questions?

Here are some related questions:

1. How do you convert the given fractions to have a denominator of 42?
2. What is the LCM of two numbers, such as 14 and 21?
3. How do you simplify fractions?
4. What is the difference between LCM and GCD?
5. How can you add fractions with different denominators?

Tip: When finding the LCM, always use the highest powers of each prime number.