Understanding Least Common Multiple for Kids

To find the least common multiple (LCM) of 4 and 6, we identify the multiples of each number. The multiples of 4 are 4, 8, 12, 16, etc., while the multiples of 6 are 6, 12, 18, 24, etc. The smallest multiple that appears in both lists is 12. Therefore, 12 is the least common multiple of 4 and 6, as it is the smallest number that both 4 and 6 can divide evenly into.

The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. For 5 and 10, the multiples of 5 are 5, 10, 15, 20, etc., and the multiples of 10 are 10, 20, 30, etc. The smallest common multiple in both lists is 10, making it the LCM of 5 and 10.

Listing multiples is an effective method to find the least common multiple (LCM) of two or more numbers. This involves writing out the multiples of each number until a common multiple is identified. The smallest of these common multiples is the LCM. This method is particularly useful for smaller numbers, as it allows for a straightforward visual comparison of the multiples, making it easier to determine the least common one. Other methods, like prime factorization or division, can also be used, but listing multiples is often the simplest for beginners.

To find the least common multiple (LCM) of two numbers, we identify the smallest multiple that both numbers share. For 3, the multiples are 3, 6, 9, 12, 15, 18, 21, and for 7, the multiples are 7, 14, 21, 28. The smallest multiple that appears in both lists is 21. Thus, the LCM of 3 and 7 is 21, as it is the first number that both can divide without leaving a remainder.

To find the smallest positive integer that is a multiple of both 8 and 12, we determine their least common multiple (LCM). The prime factorization of 8 is \(2^3\) and for 12, it is \(2^2 \times 3^1\). The LCM is found by taking the highest power of each prime: \(2^3\) from 8 and \(3^1\) from 12. This gives us \(2^3 \times 3^1 = 8 \times 3 = 24\). Thus, 24 is the smallest number that is a multiple of both 8 and 12.

To find the first multiple of 9 that is also a multiple of 12, we can list the multiples of each. The multiples of 9 are 9, 18, 27, 36, 45, etc., while the multiples of 12 are 12, 24, 36, 48, etc. The first number that appears in both lists is 36. This means that 36 is the smallest number that is divisible by both 9 and 12, confirming it as the first common multiple.

The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. For 2 and 5, the multiples of 2 are 2, 4, 6, 8, 10, and the multiples of 5 are 5, 10, 15, 20. The smallest multiple that appears in both lists is 10. Therefore, the LCM of 2 and 5 is 10.

Prime factorization is a method for finding the least common multiple (LCM) by breaking down each number into its prime factors. Once the prime factors are identified, the LCM is determined by taking the highest power of each prime factor that appears in the factorization of all numbers involved. This approach ensures that all multiples are accounted for, making it a systematic and reliable method for calculating the LCM. Other methods like multiplication, subtraction, or division do not provide a comprehensive way to determine the LCM.

To find the least common multiple (LCM) of 6 and 15, we identify the multiples of both numbers. The multiples of 6 are 6, 12, 18, 24, 30, and those of 15 are 15, 30, 45, and so on. The smallest common multiple in both lists is 30. Additionally, using the prime factorization method, 6 can be expressed as 2 × 3 and 15 as 3 × 5. The LCM is obtained by taking the highest powers of all prime factors: 2^1, 3^1, and 5^1, resulting in 2 × 3 × 5 = 30.

The Least Common Multiple (LCM) of 1 and any number is the number itself because the LCM is the smallest multiple that two numbers share. Since 1 is a factor of every integer, the multiples of 1 include all integers, making the smallest common multiple between 1 and any number equal to that number. Therefore, for any given number, it is the least multiple that can be formed with 1.

To find the least common multiple (LCM) of 10 and 20, we identify the multiples of each number. The multiples of 10 are 10, 20, 30, 40, 50, while the multiples of 20 are 20, 40, 60. The smallest multiple that appears in both lists is 20. However, since the answer given is 40, it seems there may be a misunderstanding. The actual LCM of 10 and 20 is 20, as 20 is the smallest number that both can divide without a remainder. Therefore, the correct answer should be 20, not 40.

not-available-via-ai

To find the least common multiple (LCM) of 3, 4, and 5, we identify the highest powers of their prime factors. The prime factorization of 3 is 3, for 4 it is \(2^2\), and for 5 it is 5. The LCM is obtained by multiplying these highest powers together: \(2^2 \times 3^1 \times 5^1 = 4 \times 3 \times 5 = 60\). Thus, 60 is the smallest number that is a multiple of all three numbers.

Adding numbers is not a method to find the least common multiple (LCM) because it does not involve any systematic approach to identify multiples of the given numbers. The LCM is determined by methods that focus on the multiples or factors of the numbers, such as the ladder method, listing multiples, and prime factorization. These techniques specifically help in finding the smallest common multiple, while simply adding the numbers does not provide any relevant information about their multiples.

To find the least common multiple (LCM) of 8 and 10, we first identify the prime factorization of each number. The prime factors of 8 are \(2^3\) and for 10, they are \(2^1 \times 5^1\). The LCM is determined by taking the highest power of each prime factor present in either number. Here, the highest power of 2 is \(2^3\) and for 5, it is \(5^1\). Therefore, the LCM is \(2^3 \times 5^1 = 8 \times 5 = 40\).

Multiples of a number are obtained by multiplying that number by integers. For the number 7, the sequence of multiples starts with 7 (7x1), followed by 14 (7x2), and then 21 (7x3). The next integer in this sequence is 4, so the next multiple is 7 multiplied by 4, which equals 28. Thus, following the established pattern of adding 7, the next multiple of 7 after 21 is indeed 28.

To find the least common multiple (LCM) of 12 and 15, we first determine their prime factorizations: 12 = 2² × 3 and 15 = 3 × 5. The LCM is found by taking the highest power of each prime factor present in the factorizations. Thus, we take 2² (from 12), 3¹ (common to both), and 5¹ (from 15). Multiplying these together gives us LCM = 2² × 3¹ × 5¹ = 4 × 3 × 5 = 60. Therefore, the LCM of 12 and 15 is 60.

The smallest number that is a multiple of both 2 and 3 is their least common multiple (LCM). To find the LCM of 2 and 3, we consider the multiples of each: the multiples of 2 are 2, 4, 6, 8, and so on, while the multiples of 3 are 3, 6, 9, 12, etc. The smallest common multiple between these two lists is 6. Therefore, 6 is the smallest number that satisfies being a multiple of both 2 and 3.

To find the least common multiple (LCM) of 9 and 12, we first determine the prime factorization of each number. The prime factors of 9 are \(3^2\) and for 12 are \(2^2 \times 3\). The LCM is found by taking the highest power of each prime factor present in both factorizations. Thus, we take \(2^2\) from 12 and \(3^2\) from 9. Multiplying these together gives \(2^2 \times 3^2 = 4 \times 9 = 36\). Therefore, the LCM of 9 and 12 is 36.

The least common multiple (LCM) of two numbers is the smallest multiple that is evenly divisible by both numbers. In this case, since both numbers are 1, the only multiple of 1 is 1 itself. Therefore, the LCM of 1 and 1 is 1, as it is the smallest number that both can divide without leaving a remainder.