Question

List all the common multiples of \(2,3,\) and 15 that are less than 100 .

Short answer

The common multiples of 2, 3, and 15 that are less than 100 are 30, 60, and 90.

Step-by-step solution

- Identify the Least Common Multiple (LCM) of the Numbers

Find the LCM of the given numbers: 2, 3, and 15. The LCM is the smallest number that is a multiple of each of the given numbers.

- Find the Prime Factorization

Prime factorize each number:2: \(2eq\textprime\factor\)3: \(3eq\textprime\factor\)15: \(3 \times 5eq\textprime\factor\)

- Calculate the LCM Using Prime Factors

Multiply the highest power of each prime factor present in the factorizations. The highest powers are 2, 3, and 5, which gives the LCM: \(2 \times 3 \times 5 = 30\)

- List Multiples of the LCM That Are Less Than 100

List the multiples of 30 that are less than 100. These multiples are: 30, 60, 90.

- Verify the Multiples

Ensure that each of the listed multiples (30, 60, and 90) is indeed a multiple of 2, 3, and 15:- 30: \(30\triangleleft\)`\times`2, \`=\`- 30: \(30\div\tripspan:3\to\triangleright\toatimes\thenewline=\backslashdividedby =5- n0\=30/15\) \

Key concepts

least common multiple

The Least Common Multiple (LCM) of a set of numbers is the smallest number that is a multiple of each number in the set. It's important because it helps you find a number that evenly divides by all the given numbers. In this exercise, we need to find the LCM of 2, 3, and 15. The key step is to consider the prime factors of each number and then take the highest powers of each prime factor to calculate the LCM. When we do this, we find the LCM is 30. This means 30 is the smallest number that all three numbers (2, 3, and 15) can divide evenly.

prime factorization

Prime factorization is breaking down a number into its basic building blocks, which are prime numbers. For example, the prime factorization of 2 is just 2 because 2 is a prime number. For 3, it's also just 3. For 15, we break it down into two prime numbers, 3 and 5, so its prime factorization is 3 × 5. This step is crucial in finding the LCM because it allows us to see which prime numbers are involved and how many times each prime factor appears. By this method, we identify that our LCM for the numbers 2, 3, and 15 will involve 2, 3, and 5.

multiples

Multiples of a number are the results you get when you multiply that number by the integers (1, 2, 3, etc.). For instance, the multiples of 2 are 2, 4, 6, 8, and so on. In this exercise, once we find the LCM of 2, 3, and 15, we then list the multiples of this LCM that are less than 100. Since the LCM is 30, the multiples of 30 below 100 are 30, 60, and 90. These are the numbers that are common multiples of 2, 3, and 15 and are below 100. We verify these by ensuring each number can be divided evenly by 2, 3, and 15.