Even numbers
Any number that can be divided by \({2}\) is called an even number.
Some examples of even numbers are \({2}\), \({4}\), \({6}\), \({8}\), \({10}\), \({22}\), \({144}\) and \({2,020}\).
If a number ends with an even digit, then it is an even number. For example \({1~023~458}\) is even, because the final digit (\({8}\)) is even.
Odd numbers
Any number that cannot be divided by \({2}\) is called an odd number.
Some examples of odd numbers are \({1}\), \({3}\), \({5}\), \({7}\), \({9}\), \({35}\), \({177}\), \({2,435}\), etc.
If a number ends with an odd digit, then it is an odd number.
For example \({3~702~443}\) is odd, because the final digit (\({3}\)) is odd.
Square numbers
Square numbers are formed by multiplying a number by itself.
The first four square numbers are:
\(1 \times 1 = 1\)
\(2 \times 2 = 4\)
\(3 \times 3 = 9\)
\(4 \times 4 = 16\)
A shorthand way of writing a square number is to use the power \({2}\).
For example, \({3}\times{3}\) can be written as \({3}^{2}\).
Each square number can be represented as a square of dots:
Question
Write down the next six square numbers (from \({5}\) onwards). Then check your answer.
Answer
\(5 \times 5 = 25\), \(6 \times 6 = 36\), etc.
So the next square numbers are:
\(25\), \(36\), \(49\), \(64\), \(81\) and \(100\)
Cube numbers
Cube numbers are formed by multiplying a digit by itself three times.
For example, \(1\) cubed is \(1 \times 1 \times 1 = 1\)
\(2\) cubed is \(2 \times 2 \times 2 = 8\)
\(3\) cubed is \(3 \times 3 \times 3 = 27\)
\(4\) cubed is \(4 \times 4 \times 4 = 64\)
A shorthand way of writing a cube number is to use the power \({3}\).
For example, \({4}\times{4}\times{4}\) can be written as \({4}^{3}\).
Each cube number can be represented by a cube made up of unit cubes:
Question
Use a calculator to find the next six cube numbers (from \({5}\) onwards) and then check your answer.
Answer
\(5 \times 5 \times 5 = 125\), \(6 \times 6 \times 6 = 216\), etc.
So the next six cube numbers are:
\(125\), \(216\), \(343\), \(512\), \(729\) and \({1,000}\)
Triangle numbers
Triangle numbers can be represented as a triangle of dots.
The triangle numbers are:
Question
What is the next triangle number after \(21\)?
Answer
The next row of the triangle will have \(7\) dots.
So the next number is:
\(1 + 2 + 3 + 4 + 5 + 6 + 7 = 28\)
Multiples
The multiples of a number are those numbers that it will divide into exactly.
For example, the multiples of \({5}\) are \({5}\), \({10}\), \({15}\), \({20}\), \({25}\), \({30}\), \({…}\)
Remember - multiples are like times tables:
\({1}\times{5} = {5}\)
\({2}\times{5} = {10}\)
\({3}\times{5} = {15}\)
\({4}\times{5} = {20}\)
Therefore, the multiples of \(5\) are \(5\), \(10\), \(15\), \(20\), \({…}\)
Similarly the multiples of \({7}\) are \({7}\), \({14}\), \({21}\), \({28}\), \({35}\), \({42}\), \({…}\)
Question
What are the first five multiples of \(11\)?
Answer
\(11\), \(22\), \(33\), \(44\), and \(55\).
Remember that the first multiple of any number is always the number itself.
Test section
Question 1
Which of these is not an odd number?
a) \({17}\)
b) \({3,652,331,666,549}\)
c) \({574}\)
Answer
\(574\)
Question 2
What is the sixth even number?
a) \({12}\)
b) \({6}\)
c) \({10}\)
Answer
The sixth even number is \({6}\times{2}={12}\)
Question 3
Which of these is a square number?
a) \({32}\)
b) \({144}\)
c) \({40}\)
d) \({2}\)
Answer
\({144}={12}\times{12}\), and so it's a square number.
Question 4
Which of these is a cube number?
a) \({1}\)
b) \({6}\)
c) \({9}\)
Answer
\({1}\times{1}\times{1}={1}\), so \({1}\) is a cube number.
Question 5
Which of these is not a cube number?
a) \({64}\)
b) \({99}\)
c) \({1,000}\)
Answer
\({99}\) isn't equal to any whole number cubed.
Question 6
Which of these is a triangle number?
a) \({15}\)
b) \({20}\)
c) \({25}\)
Answer
The correct answer is a).
\(1+2+3+4+5=15\)
Question 7
Which of these is a square number and a triangle number?
a) \({55}\)
b) \({100}\)
c) \({36}\)
Answer
\(36=6\times6\), so it's a square number; and \(36=1+2+3+4+5+6\), so it's a triangle number.
Question 8
What is the \({99}^{th}\) multiple of \({7}\)?
a) \({997}\)
b) \({693}\)
c) \({630}\)
Answer
The correct answer is b).
How did you get to the answer?
One method is to multiply \({7}\times{100}\) and subtract \({7}\).
Question 9
Which of the following numbers is a square number which is odd?
a) \({36}\)
b) \({111}\)
c) \({81}\)
Answer
\({81}\).
Question 10
What's the first multiple of \({8}\) which is also a multiple of \({6}\)?
a) \({12}\)
b) \({48}\)
c) \({24}\)
Answer
\({24}\) is a multiple of both \({8}\) and \({6}\), and is the lowest value that fits that description.
More on Number patterns
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