Numbers greater than 1 with only a trivial rectangular array are called prime numbers. All other whole numbers greater than 1 are called composite numbers. We shall discuss prime numbers in a great deal more detail in the later module, Primes and Prime Factorisation.
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The usual definition of a prime number expresses exactly the same thing in terms of factors:. Here are the only possible rectangular arrays for the first four prime numbers:. Rectangular arrays are not the only way that numbers can usefully be represented by patterns of dots. In the diagram to the right, two copies of the fourth triangular number have been fitted together to make a rectangle. Explain how to calculate from this diagram that the 4th triangular number is Hence calculate the th triangular number.
Multiples, common multiples and the LCM. Because 6 is an even number, all its multiples are even. The multiples of an odd number such as 7, however, alternate even, odd, even, odd, even,… because we are adding the odd number 7 at each step. The number zero is a multiple of every number. The multiples of zero are all zero.
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Every other whole number has infinitely many multiples. We can illustrate the multiples of a number using arrays with three columns and an increasing numbers of rows. Here are the first few multiples of Rows and columns can be exchanged. Thus the multiples of 3 could also be illustrated using arrays with three rows and an increasing numbers of columns. The repeating pattern of common multiples is a great help in understanding division. Here again are the multiples of 6,. If we divide any of these multiples by 6, we get a quotient with remainder zero.
For example,. To divide any other number such as 29 by 6, we first locate 29 between two multiples of 6. Thus we locate 29 between 24 and Notice that the remainder is always a whole number less than 6, because the multiples of 6 step up by 6 each time. Hence with division by 6 there are only 6 possible remainders,. This result takes a very simple form when we divide by 2, because the only possible remainders are 0 and 1.
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. The table above shows all the whole numbers written out systematically in 7 columns. Suppose that each number in the table is divided by 7 to produced a quotient and a remainder.
When 22 and 41 are divided by 6, their remainders are 4 and 5 respectively. An important way to compare two numbers is to compare their lists of multiples. Let us write out the first few multiples of 4, and the first few multiples of 6, and compare the two lists. The numbers that occur on both lists have been circled, and are called common multiples. Apart from zero, which is a common multiple of any two numbers, the lowest common multiple of 4 and 6 is These same procedures can be done with any set of two or more non-zero whole numbers.
Two or more nonzero numbers always have a common multiple — just multiply the numbers together.
But the product of the numbers is not necessarily their lowest common multiple. You will have noticed that the list of common multiples of 4 and 6 is actually a list of multiples of their LCM Similarly, the list of common multiples of 12 and 16 is a list of the multiples of their LCM This is a general result, which in Year 7 is best demonstrated by examples.
In an exercise at the end of the module, Primes and Prime Factorisation , however, we have indicated how to prove the result using prime factorisation. Factors, common factors and the HCF. The number zero is a multiple of every number, so every number is a factor of zero. On the other hand, zero is the only multiple of zero, so zero is a factor of no numbers except zero. These rather odd remarks are better left unsaid, unless students insist. They should certainly not become a distraction from the nonzero whole numbers that we want to discuss.
The product of two nonzero whole numbers is always greater than or equal to each factor in the product.
Hence the factors of a nonzero number like 12 are all less than or equal to Thus whereas a positive whole number has infinitely many multiples, it has only finitely many factors. The long way to find all the factors of 12 is to test systematically all the whole numbers less than 12 to see whether or not they go into 12 without remainder. The list of factors of 12 is.
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It is very easy to overlook factors by this method, however. A far more efficient way, is to look for pairs of factors whose product is Begin by testing all the whole numbers 1, 2, … that could be the smaller of a pair of factors with product 12,. We can display these pairs of factors by writing the 12 dots in all possible rectangular arrays:. For a larger number such as 60, the method recommended here has the following steps:. Another important way to compare two numbers is to compare their lists of factors. Let us write out the lists of factors of 18 and 30, and compare the lists.
The numbers that occur on both lists have been circled, and are called the common factors. As with common multiples, these procedures can be done with any list of two or more whole numbers. Any collection of whole numbers always has 1 as a common factor. The question is whether the numbers have common factors greater than 1. 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 A nonzero whole number has only a finite number of factors, so it has a greatest factor. Two or more numbers always have a HCF because at least one of them is nonzero.
These are distractions from the main ideas. You will have noticed that the list of common factors of 18 and 30 is actually a list of factors of their HCF 6. Similarly, the list of common factors of 30 and 75 is a list of the factors of their HCF Again, this is a general result, which in Year 7 is best demonstrated by examples. An exercise at the end of the module Primes and Prime Factorisation indicates how to prove the result using prime factorisation.
The two relationships below between the HCF and the LCM are again best illustrated by examples in Year 7, but an exercise in the module Primes and Prime Factorisation indicates how they can be proven. The first relationship is extremely useful, and is used routinely when working with common denominators of fractions. The second relationship is not so obvious, and needs to be brought out by examples.
The three-part example below indicates how this relationship can be proven from the relationship above, although such a proof would be unsuited for most Year 7 students. As we saw earlier in the module, we can arrange 3 2 dots in a square and 3 3 dots in a cube:. As mentioned before, there is no straightforward geometrical representation of higher powers.
Powers of numbers are used extensively later in the study of logarithms and of combinatorics. It is useful to be able to compute or remember some smaller powers quickly, and recognise them. The powers of 2 are: 2, 4, 8, 16, 32, 64, , , , , , , ,….
Fun with digits
Our base 10 place-value system displays every number as a sum of multiples of powers of Computers use a base 2 place-value system, so computer programmers need to know the powers of 2, and must be able to convert numbers quickly to a sum of powers of 2. There are several straightforward tests for divisibility that are very useful when factoring numbers.
They all have their origin in the base 10 that we use for our system of numerals. Because 10 is a multiple of 2, every multiple of 10 is a multiple of 2. Thus to test whether a number is divisible by 2, we only need to look at the last digit. Similarly, 10 is a multiple of 5, so every multiple of 10 is a multiple of 5. Thus to test whether a number is divisible by 5, we only need to look at the last digit. Thus to test for divisibility by 4 and 25, we only need to look at the last two digits.
Old school equations with Sal
The tests for divisibility by 9, and in turn by 3, arise from the fact that 9 is 1 less than The general result is. Because 9 is a multiple of 3, the remainders after division by 3 follows a similar pattern,. Hence 62 has remainder 6 when divided by 9, and remainder 0 when divided by 3.
It was important here to write the alternating sum of the digits starting at the right-hand end with the units. The general result is,. We can combine the previous tests for divisibility by testing separately for divisibility by the highest power of each prime.
Here are examples of some such tests:. Divisibility by powers of 10 is particularly simple—just count the number of zeroes. The multiplication table is one of the best-known objects in arithmetic. It is formed by doing nothing more that writing out the first twelve non-zero multiples of each number from 1 to 12 in twelve successive rows. Or perhaps they were written out in twelve successive columns. Despite the simplicity of its construction, it is a very powerful object indeed, and well justifies the recommendation to learn it by heart.
Here are some of its many properties — students routinely find many more. Students often see many more patterns in this table. The following exercise gives some less obvious properties, but the proofs are omitted, because they require quite serious algebra. Once sequences and series are being studied, perhaps in Year 11, the table is well worth revisiting because of the insights it can give into sequences and into the use of algebra.
There are five useful laws, collectively called the index laws, that help us manipulate powers.