A nonzero whole number has only a finite number of factors, so it has a greatest factor. The only numbers in the module are whole numbers, apart from the final paragraphs, where fractions are used so that the fifth index law can be presented in a more satisfactory form.
There are several groups of well-known divisibility tests that can check whether a number is a factor without actually performing the division.
Any collection of whole numbers always has 1 as a common factor. In later years, when students have become far more confident with algebra, these remarks about division can be written down very precisely in what is called the division algorithm.
Which of these numbers are prime, which are composite, which are square, and which are even?
Exercise 17 [A rather difficult challenge activity] Adding successive cubes gives the squares of the triangular numbers: There are five useful laws, collectively called the index laws, that help us manipulate powers.
The array representing the even number 10 has the dots divided evenly into two equal rows of 5, but the array representing the odd number 11 has an extra odd dot left over. The question is whether the numbers have common factors greater than 1.
The following exercise gives some less obvious properties, but the proofs are omitted, because they require quite serious algebra. It is very easy to overlook factors by this method, however. Solution a The LCM of 9 and 10 is their product 90.
For example, the arrays below illustrate significant properties of the numbers 10, 9, 8 and 7. Draw four diagrams to illustrate the four cases of subtraction of odd and even numbers. I am a nerd and have been living and breathing computers ever since 10th grade back in 1981.
An important way to compare two numbers is to compare their lists of multiples. I mean, you are looking at a computer monitor right now. The numbers that occur on both lists have been circled, and are called common multiples. Example Test for divisibility by 11: So a bit represents values from 0 to 1 2 values.
The usual definition of a prime number expresses exactly the same thing in terms of factors:. The general result is. The other cases are very similar. Factoring by taking out the HCF is first considered in the module Algebraic Expressions , and developed further in the module, Negatives and the Index Laws.
For example, when we look at 30 and 12, we see that they are both multiples of 6, and that 6 is the greatest factor common to both numbers. They all have their origin in the base 10 that we use for our system of numerals. Every whole number is a factor of 0, so the common factors of 0 and say 12 are just the factors of 12, and the HCF of 0 and 12 is 12.