What is the least common multiple (LCM) of the denominators for the fractions 16 and 58?

The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. To find the LCM of the denominators 16 and 58, we can use the following steps:

1. Prime Factorization: First, we will find the prime factorization of both numbers.
2. • For 16:
     • 16 = 2 × 2 × 2 × 2 = 2⁴
   • For 58:
     • 58 = 2 × 29
3. Select Maximum Powers: Next, we take each prime factor and use its highest power from the factorizations:
4. • For the prime factor 2, the maximum power is 2⁴ (from 16).
   • For the prime factor 29, the maximum power is 29¹ (from 58).
5. Multiply the Results: Now we multiply these maximum powers together to get the LCM:
6. LCM = 2⁴ × 29¹ = 16 × 29 = 464

Thus, the least common multiple of the denominators 16 and 58 is 464.