Whether a number is a multiple of another number is an important concept in mathematics. Determining if one number is a multiple of another allows us to understand factors, division, and patterns in numbers. In this article, we will explore the question of whether 42 is a multiple of 4.

## What is a Multiple?

A multiple is a number that can be divided evenly by another number without any remainder. For example, 10 is a multiple of 5 because 10 divided by 5 equals 2 with no remainder. The number 5 is called a factor of 10. Every multiple of a number is also divisible by that number.

Some key characteristics of multiples:

- 0 is a multiple of every number
- A number is always a multiple of itself
- Even numbers are multiples of 2
- Multiples can continue infinitely (e.g. 10, 20, 30 are multiples of 10)

Determining if a number is a multiple of another is simple: divide the first number by the second. If the division results in a whole number with no remainder, then the first number is a multiple of the second.

## Is 42 a Multiple of 4?

Now let’s examine our specific question – is 42 a multiple of 4?

To find out, we divide 42 by 4:

42 / 4 = 10 remainder 2

Because there is a remainder of 2 left over, 42 is not a multiple of 4. For 42 to be a multiple of 4, the division would need to result in a whole number with no remainder.

## Examples of Multiples of 4

While 42 is not a multiple of 4, there are many other numbers that are multiples of 4. Here are some examples:

- 4
- 8
- 12
- 16
- 20
- 24
- 28
- 32
- 36
- 40
- 44
- 48

Notice that every number on this list is divisible by 4 with no remainder. We can keep listing multiples of 4 infinitely – 52, 56, 60, 64 and so on.

## Finding Multiples of a Number

While we can test division to see if a number is a multiple, there are also some shortcuts we can use:

- Multiples of a number will end in 0, 2, 4, 6, or 8
- Even numbers are always multiples of 2
- Numbers ending in 0 or 5 are always multiples of 5

We can also find multiples by counting up in increments of the factor number. For example, to generate multiples of 4 we can start at 0 and count up by 4s – 0, 4, 8, 12, 16, 20 etc. This pattern will produce the multiples of 4.

## Common Multiples

Two or more numbers can share a common multiple when they have the same factor. The smallest number that is a multiple of two or more numbers is called the least common multiple (LCM). For example:

- Multiples of 2: 2, 4, 6, 8, 10
- Multiples of 3: 3, 6, 9, 12, 15

The least common multiple of 2 and 3 is 6. This is the smallest number that is divisible by both 2 and 3. Understanding least common multiples is useful for solving math problems involving fractions.

## Real World Uses of Multiples

Identifying multiples has many practical applications including:

**Scheduling**– Finding time slots based on multiples of hours or days**Data transmission**– Sending data in blocks that are multiples of packet sizes**Fractions**– Converting between fractions and decimals using least common multiples**Measurement**– Converting between units using conversion factors which are multiples**Even distribution**– Dividing items fairly by spreading into equal groups based on multiples

## Multiples in Computer Science

In computer science, multiples play an important role in:

**Memory addresses**– Memory is allocated in blocks based on powers of 2**Data alignment**– Data structured in multiples of word sizes for optimization**Cryptography**– Security depends on finding very large prime numbers and their multiples**Error detection**– Checksums used to detect errors are based on modulo arithmetic of multiples

Understanding multiples allows optimizing performance, finding patterns, and creating efficient systems.

## Summary

In summary:

- A multiple is a number that can be divided evenly by another number without remainder
- To test if a number is a multiple, divide it by the factor and check for a whole number result
- 42 is not a multiple of 4 because 42/4 results in a remainder of 2
- Examples of multiples of 4 are 4, 8, 12, 16 and so on
- Multiples have many practical applications in scheduling, data transmission, fractions, measurement, distribution, computer science and more

Checking whether numbers are multiples of each other is an essential mathematical skill with wide utility across many fields and applications. Understanding factors, division, patterns and distribution depends on a solid grasp of multiples. While 42 is not a multiple of 4, examining their relationship provides insight into the underlying concepts.

## Conclusion

Through examples and analysis we have determined that 42 is not a multiple of 4. But exploring this concept leads to a deeper understanding of important mathematical properties of multiples that are critical for many calculations and systems. Comprehending the intricate connections between numbers is key to mastering broader math skills and thinking critically.