Isabel Must Choose A Number Between 67 And 113 That Is A Multiple Of 2, 8, And 16. Write All The Numbers That She Could Choose. If There Is More Than One Number, Separate Them With Commas.

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Introduction

In mathematics, finding the common multiple of a set of numbers is a fundamental concept that has numerous applications in various fields. In this article, we will delve into the world of multiples and explore the problem of finding a number between 67 and 113 that is a multiple of 2, 8, and 16. We will break down the problem step by step and provide a comprehensive solution.

Understanding Multiples

A multiple of a number is the product of that number and an integer. For example, the multiples of 2 are 2, 4, 6, 8, 10, and so on. Similarly, the multiples of 8 are 8, 16, 24, 32, and so on. To find the common multiple of two or more numbers, we need to find the smallest number that is a multiple of all the given numbers.

Finding the Least Common Multiple (LCM)

The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers. To find the LCM, we can use the following steps:

  1. List the multiples of each number.
  2. Identify the smallest number that is a multiple of all the given numbers.

Finding the LCM of 2, 8, and 16

To find the LCM of 2, 8, and 16, we can list the multiples of each number and identify the smallest number that is a multiple of all three numbers.

  • Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 100, 102, 104, 106, 108, 110, 112, 114, ...
  • Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104, 112, ...
  • Multiples of 16: 16, 32, 48, 64, 80, 96, 112, ...

From the above lists, we can see that the smallest number that is a multiple of all three numbers is 16.

Finding the Numbers between 67 and 113

Now that we have found the LCM of 2, 8, and 16, we need to find the numbers between 67 and 113 that are multiples of 16.

  • Multiples of 16 between 67 and 113: 80, 96, 112

Introduction

In our previous article, we explored the problem of finding a number between 67 and 113 that is a multiple of 2, 8, and 16. We found that the least common multiple (LCM) of 2, 8, and 16 is 16, and the numbers between 67 and 113 that are multiples of 16 are 80, 96, and 112. In this article, we will answer some frequently asked questions related to finding the common multiple.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of all the given numbers.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you can use the following steps:

  1. List the multiples of each number.
  2. Identify the smallest number that is a multiple of all the given numbers.

Q: What is the difference between the LCM and the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) of two or more numbers is the largest number that divides all the given numbers without leaving a remainder. The LCM and GCD are related by the following formula:

LCM(a, b) × GCD(a, b) = a × b

Q: How do I find the GCD of two or more numbers?

A: To find the GCD of two or more numbers, you can use the following steps:

  1. List the factors of each number.
  2. Identify the largest number that is a factor of all the given numbers.

Q: What is the relationship between the LCM and the GCD?

A: The LCM and GCD are related by the following formula:

LCM(a, b) × GCD(a, b) = a × b

This formula shows that the product of the LCM and GCD of two numbers is equal to the product of the two numbers.

Q: How do I use the LCM and GCD in real-life situations?

A: The LCM and GCD have numerous applications in real-life situations, such as:

  • Music: The LCM and GCD are used to find the common time signature and key of a piece of music.
  • Cooking: The LCM and GCD are used to find the common ingredient and cooking time of a recipe.
  • Engineering: The LCM and GCD are used to find the common material and design of a structure.

Q: What are some common mistakes to avoid when finding the LCM and GCD?

A: Some common mistakes to avoid when finding the LCM and GCD include:

  • Not listing all the multiples of each number.
  • Not identifying the smallest number that is a multiple of all the given numbers.
  • Not using the correct formula to find the LCM and GCD.

Conclusion

In conclusion, finding the common multiple is a fundamental concept in mathematics that has numerous applications in various fields. By understanding the LCM and GCD, you can solve problems and make decisions in real-life situations. Remember to avoid common mistakes and use the correct formula to find the LCM and GCD.