# Multiples of 24

Twenty-four is a composite number and the first number that satisfies the form 2^{3}*q *where *q* is a prime number. It is also the third nonagonal number – a figurate number that extends the concept of triangle and square numbers to nonagons.

Twenty-four is a significant number because there are 24 hours in a day, 24 cycles in the Chinese solar year, and 24 karats in pure gold.

There’s a lot more reason as to why 24 is such an important number, but in our today’s adventure, we will journey to the world of 24 and its multiples! Are you excited for a new and fun learning lesson?

## Multiples of 24 are -48, -24, 0, 24, 48, 72, 96

Multiples of a number are formed by multiplying one integer by another. Assume that the numbers 6, 12, 18, 24, and 30 are all multiples of 6 and can be generated by simply multiplying 6 by the integers 1 to 5.

In a nutshell, multiples of 24 are the results of multiplying 24 by any integer – whether negative or positive. Hence, if a number can be expressed as 24n, where n is any integer, then it is a multiple of 24.

Now, let’s take a look at the sequence of numbers below.**-48, -24, 0, 24, 48, 72, 96**

What can you notice? In the given sequence of numbers, we can notice that the difference between any two consecutive numbers is always 24. Hence, we can say that those numbers are multiples of 24. We can also see that a negative number can also be a multiple of 24.

But what about the sequence of numbers 4, 28, 52?

Is it also a sequence of numbers that are multiples of 24? Even though the difference between any consecutive numbers is 24, we cannot say that those numbers are multiples of 24 because we cannot express it in the form 24n.

Now, let’s work on determining any multiples of 24!

## How to find the multiples of 24?

Now that we’ve properly defined multiples of 24. It is now time for us to figure out how to find any possible multiples of 24. There are two ways to determine multiples of 24: repeated addition and multiplication.

**Repeated addition** is done by adding 24 as many times as necessary. Consider the diagram below to understand the process of repeated addition.

The first and smallest multiple of any number is the number itself. Hence, the first multiple of 24 is 24.

Now, to find the 2^{nd} multiple of 24 by repeated addition, we will add 24 twice. Thus, $$24\;+\;24\;=\;48$$.

Furthermore, if we are looking for the 5^{th} multiple of 24, we will simply add 24 five times. Hence, $$24\;+\;24\;+\;24\;+\;24\;+\;24\;=\;120$$.

It is not that hard, isn’t it?

Now, how about finding the 35^{th} multiple of 24? Can we still do it using repeated addition? Of course! However, it will take us some time to arrive at an answer. Hence, the need to learn the second method – which is **multiplication**!

So.. if we are looking for the 35^{th} multiple of 24, it will take us some time to find it. Hence, getting the product of 24 and 35 is a much easier way to determine it. Thus, $$24\;\times\;35\;=\;840$$. Therefore, the 35^{th} multiple of 24 is 840.

See… it’s a more simpler method, isn’t it?

Now, let’s try to determine the 124^{th} multiple of 24, shall we?

To get the 124^{th} multiple of 24, we will simply multiply 24 by 124. Hence, $$24\;\times\;124\;=\;2,976$$. Therefore, the 124^{th} multiple of 24 is 2,976.

Let’s look at the table below. The table shows the first five multiples of 24 using the repeated addition and multiplication methods.

n^{th} Multiple | Repeated Addition | Multiplication |
---|---|---|

1^{st} multiple | 24 | 24 × 1 = 24 |

2^{nd} multiple | 24 + 24 = 48 | 24 × 2 = 48 |

3^{rd} multiple | 24 + 24 + 24 = 72 | 24 × 3 = 72 |

4^{th} multiple | 24 + 24 + 24 + 24 = 96 | 24 × 4 = 96 |

5^{th} multiple | 24 + 24 + 24 + 24 + 24 = 120 | 24 × 5 = 120 |

## Did you know that…

If a number is a multiple of 3 and 8… then we can conclude that it is also a multiple of 24?

Let’s try to see if this is true. Consider the number 2,352. We need to know if it is divisible by 24 by knowing if it is also a multiple of 3 and 8. Hence, let’s divide 2,352 by 3. Thus, $$2,352\;\div\;3\;=\;784$$. It is a multiple of 3!

Now, let’s check if it is also a multiple of 8. If we divide 2,352 by 8, we will get $$2,352\;\div\;8\;=\;294$$. It is also a multiple 8! Thus, we can say that it is a multiple of 24!

Now, let’s see if it is indeed true by dividing 2,352 by 24. Hence, $$2,352\;\div\;24\;=\;98$$. Since the result does not have a remainder, then we can definitely confirm that the trick is true!

Can you try to check if 3,240 is also a multiple of 24 using this trick?

## List of First 30 multiples of 24

There is an endless number possible of integers hence, listing all possible multiples of 24 is meaningless. However, we have made a list of the first 30 multiples of 24 that is generated by multiplying 24 by the integers 1 to 30.

Product of 24 and a positive counting number | Multiples of 24 |
---|---|

24 × 1 | 24 |

24 × 2 | 48 |

24 × 3 | 72 |

24 × 4 | 96 |

24 × 5 | 120 |

24 × 6 | 144 |

24 × 7 | 168 |

24 × 8 | 192 |

24 × 9 | 216 |

24 × 10 | 240 |

24 × 11 | 264 |

24 × 12 | 288 |

24 × 13 | 312 |

24 × 14 | 336 |

24 × 15 | 360 |

24 × 16 | 384 |

24 × 17 | 408 |

24 × 18 | 432 |

24 × 19 | 456 |

24 × 20 | 480 |

24 × 21 | 504 |

24 × 22 | 528 |

24 × 23 | 552 |

24 × 24 | 576 |

24 × 25 | 600 |

24 × 26 | 624 |

24 × 27 | 648 |

24 × 28 | 672 |

24 × 29 | 696 |

24 × 30 | 720 |

## Solving problems involving multiples of 24

Now, let’s try solving these two word problems that we may encounter in our daily lives.

**Problem #1**

Risnny has this habit of counting the steps she makes. From her school to her home, she counts her steps in sets of 24. How many steps does she walk if she was able to do a set of 17 twenty-fours?

In the given problem, we are asked to determine the total number of steps it takes Risnny to walk from her school to her house. It is said that she counts all the steps she makes in sets of 24. She was also able to make 17 sets of twenty-fours.

Hence, to find the total number of steps she makes, we need to find the 17^{th} multiple of 24.

We can get the 17^{th} multiple of 24 by multiplying 24 by 17. Thus, $$24\;\times\;17\;=\;408$$ steps. Therefore, it takes Risnny 408 steps to walk from her school to her house.

Is it an easy problem? Let’s try another one!

**Problem #2**

Shynna’s collection of lipsticks costs $24 each. If the total number of lipsticks she has is 54, how much did Shynna spend on her collection?

In the problem, it is stated that Shynna has 54 lipsticks that cost $24 each. Since we are asked to get the total amount of money she spent on those lipsticks, we need to get the 54^{th} multiple of 24. Hence, we need to get the product of 24 and 54. Thus,$$\$24\;\times\;54\;=\;\$1,296$$.

Therefore, Shynna spent a total **$1,296** on her lipstick collection!

We may sometimes overlook that certain problems that we encounter in our daily lives need our little knowledge about multiples. Now, let’s see if you have really understood our lesson and take these three short questions!