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Multiples of a number are the results of multiplying that number by whole numbers. So, multiples of 3 are the results of multiplying 3 by 1, then by 2, and so on. For example, 3 multiplied by 1 is 3, 3 multiplied by 2 is 6, and so forth. The multiples of 3 less than 60 are:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, and 57.

These numbers all share a common factor, which is 3, meaning they are all divisible by 3 without leaving a remainder. Listing these multiples helps in identifying common multiples with other numbers, like 5. Understanding multiples is key to solving various arithmetic problems and is useful for finding common multiples, as seen in this exercise.

Just like multiples of 3, multiples of 5 are found by multiplying 5 by whole numbers. The calculation starts with 5 multiplied by 1, then by 2, and continues. Here are the multiples of 5 less than 60:

5, 10, 15, 20, 25, 30, 35, 40, 45, 50, and 55.

In all these multiples, the final digit ends in either 0 or 5, which is a quick way to check if a number is a multiple of 5. Knowing the list of multiples of 5 is useful not just for finding common multiples but also for understanding broader concepts like divisibility rules and factors in arithmetic problems.

The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both. In this exercise, finding common multiples of 3 and 5 that are less than 60 helps illustrate this. To find common multiples, we list the multiples of each number and look for ones they share. Here are the common multiples of 3 and 5 less than 60:

15, 30, 45.

The LCM is the lowest of these shared numbers, which is 15. Calculating common multiples is fundamental to many math problems, from simplifying fractions to solving word problems involving repetitions or cycles. Understanding the concept of LCM makes it easier to solve problems involving scheduling, ratios, and more complex arithmetic.