Determining whether a number is a multiple of another number is an important concept in mathematics. In this article, we will explore why 42 is a multiple of 6.
42 is an interesting number that has been referenced many times in pop culture. You may recognize it as the “Answer to the Ultimate Question of Life, the Universe, and Everything” from Douglas Adams’ popular science fiction series The Hitchhiker’s Guide to the Galaxy. But 42 also has mathematical significance as a multiple of the number 6.
A multiple is a number that can be divided evenly by another number. For example, 12 is a multiple of 3 because 12 divided by 3 gives an integer result of 4, with no remainder. To determine if 42 is a multiple of 6, we simply need to divide 42 by 6 and see if the result is an integer.
Dividing 42 by 6
Let’s divide 42 by 6 step-by-step:
42 / 6 = 7
The result is 7, which is an integer. Therefore, 42 is evenly divisible by 6, making it a multiple.
Using Prime Factorization
We can also use prime factorization to determine if 42 is a multiple of 6. To find the prime factors of a number, we break the number down into its prime components. The prime factors of 42 are:
42 = 2 x 3 x 7
The prime factors of 6 are:
6 = 2 x 3
We can see that 6 shares the prime factors 2 and 3 with 42. This tells us that 42 must be a multiple of 6.
Using a Remainder Check
There is another easy way to check – using remainders. When you divide one number by another, if the remainder is 0, then the number is a multiple. Let’s divide 42 by 6 again and inspect the remainder:
42 / 6 = 7 with remainder 0
Since the remainder is 0, 42 is a multiple of 6.
Examples of Other Multiples of 6
Now that we’ve confirmed 42 is a multiple of 6, let’s look at some other multiples of 6:
| Multiple of 6 |
|---|
| 6 |
| 12 |
| 18 |
| 24 |
| 30 |
| 36 |
| 42 |
| 48 |
As we can see, 42 fits into the pattern of numbers that are evenly divisible by 6. All of the multiples of 6 can be divided by 6 to get an integer result.
Common Multiples
The numbers 6 and 42 also share other common multiples. These are numbers that are multiples of both 6 and 42. Some examples are:
- 126
- 252
- 378
To find common multiples, we can multiply each original number by integers until we get overlapping results like the examples above.
Least Common Multiple
In addition to common multiples, we can also find the least common multiple (LCM) of 6 and 42. The LCM is the smallest multiple that both original numbers share.
To find the LCM of 6 and 42, we first prime factorize the numbers:
6 = 2 x 3
42 = 2 x 3 x 7
The LCM is found by taking the greatest factors multiplied together. So the LCM of 6 and 42 is 2 x 3 x 7 = 42.
Proof by Multiplication
We can also prove algebraically that 42 is a multiple of 6 by showing that 6 multiplied by an integer equals 42.
Let’s multiply 6 by 7:
6 x 7 = 42
Since multiplying 6 by the integer 7 gives us 42, this proves that 42 is a multiple of 6.
Significance of Multiples
Identifying multiples and common multiples has many useful applications. Some examples include:
- Converting between units of measurement
- Simplifying fractions by finding common denominators
- Determining all possible durations for events in scheduling problems
- Scaling recipes up or down
- Finding compatible gears for machines
As we can see, multiples are an important foundation for many areas of math and science.
Multiples in Math Patterns
Multiples also create interesting number patterns. For example, the multiples of 6 form a pattern where every 2nd number is divisible by 2, and every 3rd number is divisible by 3:
6, 12, 18, 24, 30, 36, 42, 48, 54, 60…
Noticing patterns like this can help develop number sense and make it easier to identify multiples.
Checking for Validity
To double check our work, let’s review the different ways we’ve verified that 42 is a multiple of 6:
- Dividing 42 by 6 gives an integer result
- The prime factors of 6 are all factors of 42
- Dividing 42 by 6 gives a remainder of 0
- We found a common integer (7) that proves 6 x 7 = 42
Using these different methods, we have conclusively proven that 42 is indeed a multiple of 6.
Conclusion
In summary, a multiple is a number that can be divided evenly by another number. There are several ways to check if a number is a multiple, including: dividing to check for an integer result, finding common prime factors, checking for a remainder of 0, multiplying to prove equality, and looking for number patterns.
Applying these methods to 42 and 6 demonstrates conclusively that 42 is a multiple of 6. This fundamental mathematical relationship reveals the deep and meaningful connection between these two numbers, shedding insight on patterns and structures vital to math and science.
So the next time you think of 42, remember that not only is it the “Answer to the Ultimate Question of Life, the Universe, and Everything”, it is also a meaningful multiple of the number 6.