The **Multiples of 4**
are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, … i.e. M_{4} = {4, 8, 12, 16,
20, 24, 28, 32, 36, 40, …}. Multiples of 4 are the series of numbers obtained
by multiplying 4 with the natural numbers.

4 × 1 = 4

4 × 2 = 8

4 × 3 = 12

4 × 4 = 16

4 × 5 = 20

4 × 6 = 24

4 × 7 = 28

4 × 8 = 32

4 × 9 = 36

4 × 10 = 40

…

And so on.

From this series, we observe that the multiples of 4 can be
derived by multiplying 4 by natural numbers in ascending order. These multiples
form an arithmetic sequence with a common difference of 4.

**Properties of
Multiples of 4**

Multiples of 12 possess several interesting properties that make
them unique. Here are a few notable ones:

1. __Divisibility by 4__: Every multiple of 4 is divisible by 4 without leaving a
remainder. This property applies universally to all multiples of 4, as they can
be expressed as 4 multiplied by an integer.

2. __Even Numbers__: Multiples of 4 are always even numbers. An even number is any
integer that is divisible by 2. Since 4 is divisible by 2, all its multiples
will also be divisible by 2, making them even.

3. __Increasing by 4__: Each subsequent multiple of 4 is obtained by adding 4 to the
previous multiple. For instance, starting from 4, we add 4 to get 8, then 12,
and so on. This pattern continues indefinitely.

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**Patterns and
Observations:**

1. __Digit Sum__: If we observe the digit sums of multiples of 4, we notice a
pattern. The digit sums alternate between 4 and 8. For example, the digit sum
of 4 is 4, while the digit sum of 8 is 8. Similarly, the digit sum of 12 is 3
(1 + 2 = 3), and the digit sum of 16 is 7 (1 + 6 = 7). This pattern of
alternating digit sums repeats indefinitely.

2. __Repeating Last Two
Digits__: Another interesting pattern emerges
when we look at the last two digits of multiples of 4. The last two digits of
the multiples follow a cyclical pattern: 04, 08, 12, 16, 20, 24, and so on.
This cyclic nature is a result of the fact that each multiple is formed by
adding 4 to the previous multiple.

**Applications of
Multiples of 4**

The study of multiples of 4 finds applications in various
fields, including mathematics, science, and everyday life:

1. __LCM__: One of the applications of multiples of 4 is in finding the lowest
common multiple (LCM) of two or more numbers. The LCM is the lowest multiple
that two or more numbers have in common. For example, to find the LCM of 4 and 6,
we need to find the multiples of both numbers and identify the lowest multiple
they have in common. The multiples of 4 are 4, 8, 12, 16, 20, 24,
28, 32, 36, 40 …
etc. The multiples of 6 are 6, 12,
18, 24, 30, 36, 42, 48, 54, 60, … etc. The lowest multiple that they have in
common is 12. Therefore, the LCM of 4 and 6 is 12.

2. __Timekeeping__: The concept of multiples of 4 is utilized in our conventional
system of timekeeping. Hours, minutes, and seconds are divided into multiples
of 4, enabling us to track and measure time efficiently.

3. __Computing and
Programming__: In computer programming,
multiples of 4 are often used for memory allocation and addressing. Memory
addresses are usually aligned on multiples of 4 for optimal performance.

4. __Music and Rhythm__: Multiples of 4 play a crucial role in music and rhythm.
Musical beats and measures often follow patterns based on multiples of 4,
creating the foundation for various musical compositions and genres.

**Conclusion:**

The **Multiples
of 4** are the numbers obtained by multiplying 4 with natural numbers.
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32,
36, 40 … etc. The multiples of 4 have various properties, such as divisibility
by 4, even numbers, increasing by 4, digit sum, repeating last two digits, etc.
The multiples of 4 have several applications in mathematics, such as finding
the LCM, timekeeping, computing and programming, and music and rhythm.

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