**MULTIPLES**, FACTORS, AND POWERS

## ASSUMED KNOWLEDGE

- Experience with the four operations of arithmetic.
- Instant recall of the multiplication table around 12 × 12.
- Fractions and multiplication of fractions are expected only for the last of the five index laws.
- No algebra is assumed in this module.
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## MOTIVATION

Multiplication and division of whole numbers provide many surprising things. This module encourages multiplicative considering numbers and introduces ideas that are essential skills infractions and algebra.

The ideas of the module are presented in purely arithmetical form, and no algebra is employed except in some remarks that anticipate later work. The only numbers in the module are whole numbers, apart from the final paragraphs, where fractions are utilized, so your fifth index law may be presented in a far more satisfactory form.

Students first meet the distinction between odd numbers and even numbers in early primary school. Nonetheless, it pays to everywhere in mathematics. Even numbers are multiples of 2, and more generally, multiples arise throughout mathematics and everyday life. The mass of a stack of bricks is a multiple of the group of just one brick. The number of pages in a packet of notebooks is a multiple of the number of pages in one notebook.

The factors of a number may be displayed using rectangular arrays. Some numbers, such as example 30, can arise in many various ways as something,

Whereas a number such as, for example, 31 can only be written trivially as the merchandise 31 = 1 × 31. This idea results in classifying numbers as more significant than either prime or composite and listing all of the number factors.

Several categories of well-known divisibility tests will check whether a number is a factor without actually performing the division. These tests greatly simplify the listing of factors of numbers.

Repeated addition results in multiplication. Repeated multiplication in turn results in powers, and manipulating powers, in turn, relies on five index laws. Controls are introduced in this module, as well as four of the five index laws.

We are used to comparing numbers in terms of their size. The best common factor (HCF) and lowest common multiple (LCM) allow us to compare numbers in their characteristics and multiples. For instance, when we look at 30 and 12, we see that they’re both multiples of 6 and that 6 is the foremost factor common to both numbers. We also note that 60 is really a multiple of both numbers and that 60 is the cheapest common multiple (apart from 0). The HCF and LCM are essential for fractions and later for algebra.

## His pattern occurs in a variety of everyday situations:

When we walk, we step left, right, left, right,…

When music is written in double time, such as the Australian National Anthem, the notes are alternately stressed, unstressed, stressed, unstressed,…

Our time is alternately divided day, night, day, night,…

The squares on each row or column of a chessboard are alternately black, white, black, white,…

Indeed, our notion of quantity 2 is distinctive from our conceptions of most other numbers that people use in a different language. We divide a pie between two people, but among three people. We identify two alternatives but three options. The term ‘doubt relates to the Latin’ duo, ‘the phrase ‘two-faced means ‘liar, ‘and the original number of the devil is 2.

## Adding and subtracting odd and even numbers

There are several apparent facts about calculations with odd and even numbers, which are invaluable as an automatic check of calculations. First, addition and subtraction:

The sum or difference of two odd numbers, or two even numbers, is even.

The sum or difference between a peculiar number and an even number is odd.

Proofs by arrays usually convince students significantly more than algebraic proofs. The diagram below illustrates ‘odd plus odd equals even ‘and shows how everything depends upon the strange dot left over. One other case is incredibly similar.