# Decomposition of Natural Numbers (with Examples and Exercises)

The decomposition of natural numbers can occur in different ways: as a product of prime factors, as a sum of powers of two and additive decomposition. They will be explained in detail below.

A useful property that has the powers of two is that with them you can convert a decimal system number to a binary system number. For example, 7 (number in the decimal system) is equivalent to the number 111, since 7 = (2 ^ 2) + (2 ^ 1) + (2 ^ 0). The natural numbers are used to count

Natural numbers are the numbers with which you can count and list objects. In most cases, it is considered that the natural numbers start from 1. These numbers are taught at school and are useful in almost all activities of daily life.

Index

• 1 Ways to decompose natural numbers
• 1.1 Decomposition as a product of prime factors
• 1.2 Decomposition as sum of powers of 2
• 2 Exercises and solutions
• 2.1 Decomposition in product of prime numbers
• 2.2 Decomposition in sum of powers of 2
• 3 References

## Ways to decompose natural numbers

As mentioned before, here are three different ways to break down the natural numbers.

### Decomposition as a product of prime factors

Every natural number can be expressed as a product of prime numbers. If the number is already prime, its decomposition is itself multiplied by one.

If not, it is divided into the smallest prime number by which it is divisible (it can be one or several times), until a prime number is obtained.

For example:

5 = 5 * 1.

15 = 3 * 5.

28 = 2 * 2 * 7.

624 = 2 * 312 = 2 * 2 * 156 = 2 * 2 * 2 * 78 = 2 * 2 * 2 * 2 * 39 = 2 * 2 * 2 * 2 * 3 * 13.

175 = 5 * 35 = 5 * 5 * 7.

### Decomposition as sum of powers of 2

Another interesting property is that any natural number can be expressed as a sum of powers of 2. For example:

1 = 2 ^ 0.

2 = 2 ^ 1.

3 = 2 ^ 1 + 2 ^ 0.

4 = 2 ^ 2.

5 = 2 ^ 2 + 2 ^ 0.

6 = 2 ^ 2 + 2 ^ 1.

7 = 2 ^ 2 + 2 ^ 1 + 2 ^ 0.

8 = 2 ^ 3.

15 = 2 ^ 3 + 2 ^ 2 + 2 ^ 1 + 2 ^ 0.

Another way to decompose natural numbers is by considering their decimal numbering system and the positional value of each number.

This is obtained by considering the figures from right to left and starting with unit, decade, hundred, unit of a thousand, tens of thousands, hundreds of thousands, units of millions, etc. This unit is multiplied by the corresponding numbering system.

For example:

239 = 2 * 100 + 3 * 10 + 9 * 1 = 200 + 30 + 9.

4893 = 4 * 1000 + 8 * 100 + 9 * 10 + 3 * 1.

## Exercises and solutions

Consider the number 865236. Find its decomposition into the product of prime numbers, in sum of powers of 2 and its additive decomposition.

### Decomposition in product of prime numbers

-As 865236 is even, be sure that the smallest cousin for which it is divisible is 2.

-Dividing between 2 you get: 865236 = 2 * 432618. Again an even number is obtained.

-It continues dividing until an odd number is obtained. Then: 865236 = 2 * 432618 = 2 * 2 * 216309.

-The last number is odd, but it is divisible by 3 since the sum of its digits is odd.

-So, 865236 = 2 * 432618 = 2 * 2 * 216309 = 2 * 2 * 3 * 72103. The number 72103 is prime.

-Therefore the desired decomposition is the last one.

### Decomposition in sum of powers of 2

-The highest power of 2 is sought that is closest to 865236.

-This is 2 ^ 19 = 524288. Now the same thing is repeated for the difference 865236 - 524288 = 340948.

-The closest power in this case is 2 ^ 18 = 262144. It is now followed with 340948-262144 = 78804.

-In this case the nearest power is 2 ^ 16 = 65536. Continue 78804 - 65536 = 13268 and you get that the nearest power is 2 ^ 13 = 8192.

-Now with 13268 - 8192 = 5076 and you get 2 ^ 12 = 4096.

-Then with 5076 - 4096 = 980 and we have 2 ^ 9 = 512. We continue with 980 - 512 = 468, and the nearest power is 2 ^ 8 = 256.

-Now comes 468 - 256 = 212 with 2 ^ 7 = 128.

-Then, 212 - 128 = 84 with 2 ^ 6 = 64.

-Now 84 - 64 = 20 with 2 ^ 4 = 16.

-And finally 20 - 16 = 4 with 2 ^ 2 = 4.

Finally you have to:

865236 = 2 ^ 19 + 2 ^ 18 + 2 ^ 16 + 2 ^ 13 + 2 ^ 12 + 2 ^ 9 + 2 ^ 8 + 2 ^ 7 + 2 ^ 6 + 2 ^ 4 + 2 ^ 2.

Identifying the units we have that the unit corresponds to the number 6, the ten to 3, the hundred to 2, the unit of a thousand to 5, the ten thousand to 6 and the hundred thousand to 8.

Then,

865236 = 8 * 100,000 + 6 * 10,000 + 5 * 1,000 + 2 * 100 + 3 * 10 + 6

= 800,000 + 60,000 + 5,000 + 200 + 30 + 6.

## References

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