Decomposition - a MaS Tutor.

Decomposition of Numbers

The breaking up of numbers into parts either through addition (and subtraction), multiplication (and division) or both.

46 = 40 + 6

46 = 2 \times 23

46 = 6 \times 7 + 4

These methods can be useful in making calculations simpler

The breaking up of numbers by addition (and subtraction)

A = a_{1} + a_{2} + a_{3} + \dots + a_{n}

For example:

725 = 700 + 20 + 5

This can be useful when adding numbers:

725 + 483 = (700 + 20 + 5) + (400 + 80 + 3)

The above is the partitioning step and the following collects like-terms,

= 700 + 400 + 20 + 80 + 5 + 3

Now add the like-terms,

= 1100 + 100 + 8

Then completing the calculation,

= 1208

Any integer can be rewritten as a product of two of more factors

6 = 2 \times 3

210 = 21 \times 10

625 = 25 \times 25

This is a good starting point for Prime Factor Decomposition

dividend = quotient \times divisor + remainder

This type of decomposition is useful in calculations involving fractions

Follow-on study:: HCF and LCM (Care of helpwithfractions.com);; HCF and LCM (Care of thirdspacelearning.com);; Further Questions (care of corbettmaths.com); Modular Arithmetic (Care of khanacademy.org);; Partial Fractions (Care of alevelmaths.co.uk).

prime numbers: Practice Questions; Textbook Exercise; (Care of corbettmaths.com)

Every integer, $ℕ$ can be decomposed into a unique list of prime factors. This is known as the fundamental theorem of arthmetic.

Examples:

24 = 4 \times 6 = 2^{3} \times 3

210 = 6 \times 35 = 2 \times 3 \times 5 \times 7

360 = 8 \times 45 = 2^{3} \times 3^{2} \times 5

Determining the prime factor decomposition of a number can be done via a prime factor tree

∴ 56 = 2^{3} \times 7

Product of Primes
Product of Primes (squares/cubes): Practice Questions; Textbook Exercise
Product of Primes (LCM/HCF): Practice Questions; Textbook Exercise; (Care of corbettmaths.com)