Points to Remember:
- Least Common Multiple (LCM) is the smallest positive integer that is divisible by all the given numbers.
- Prime factorization is a useful method for finding the LCM.
Introduction:
The Least Common Multiple (LCM) is a fundamental concept in number theory. Given a set of integers, the LCM is the smallest positive integer that is a multiple of all the integers in the set. Finding the LCM has applications in various areas, including scheduling, fraction simplification, and solving problems related to rhythmic patterns. This response will determine the LCM of 5, 7, and 13.
Body:
Method 1: Prime Factorization
The most efficient method to find the LCM of relatively small numbers is prime factorization. Let’s find the prime factorization of each number:
- 5 = 5 (5 is a prime number)
- 7 = 7 (7 is a prime number)
- 13 = 13 (13 is a prime number)
Since 5, 7, and 13 are all prime numbers and have no common factors, the LCM is simply the product of these numbers.
Method 2: Listing Multiples
Alternatively, we can list the multiples of each number until we find the smallest common multiple:
- Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, …
- Multiples of 7: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84, 91, 98, 105, …
- Multiples of 13: 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169, 182, 195, 208, 221, 234, 247, 260, 273, 286, 299, 312, 325, 338, 351, 364, 377, 390, 403, 416, 429, 442, 455, 468, 481, 494, 507, 520, 533, 546, 559, 572, 585, 598, 601, 614, 627, 640, 653, 666, 679, 692, 705, 718, 731, 744, 757, 770, 783, 796, 809, 822, 835, 848, 861, 874, 887, 900, 913, 926, 939, 952, 965, 978, 991, 1004, 1017, 1030, 1043, 1056, 1069, 1082, 1095, …
The smallest number that appears in all three lists is 455.
Conclusion:
Both methods confirm that the Least Common Multiple (LCM) of 5, 7, and 13 is 455 (5 x 7 x 13). The prime factorization method is generally more efficient for larger numbers. Understanding LCM is crucial for various mathematical operations and problem-solving scenarios. This simple example highlights the importance of foundational mathematical concepts in building a strong understanding of more complex mathematical ideas. The ability to efficiently calculate LCM contributes to a more robust mathematical foundation.
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