Final answer:
The least common multiple of 17, 24, and 53 is 69,528.
Step-by-step explanation:
The least common multiple (LCM) of 17, 24, and 53 is easiest to find by directly multiplying the numbers together since they have no common factors other than 1 due to 17 and 53 being prime numbers and 24's prime factorization not sharing any primes with 17 or 53.
To find the least common multiple of 17, 24, and 53, we can use the prime factorization method. First, factorize each number:
- 17 = 17
- 24 = 2^3 * 3
- 53 = 53
Next, write down the highest power of each prime number that appears in any of the factorizations:
- 17 = 17^1
- 24 = 2^3 * 3^1
- 53 = 53^1
Finally, multiply the highest powers together to get the least common multiple:
LCM = 17^1 * 2^3 * 3^1 * 53^1 = 17 * 8 * 3 * 53 = 69,528