By studying this lesson, you will be able to,
● identify odd numbers, even numbers, prime numbers, composite numbers, square numbers and triangular numbers among the whole numbers and
● identify number patterns formed by the above types of numbers.
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Let us identify even numbers and odd numbers among the whole numbers.
Six pens can be equally divided between Nimali and Vimali in the following manner.
Let us see whether the number of pens in the table below can be divided equally between the two.
This shows that, if the quantity is 2, 4, 6 or 8, then it can be divided into two equal parts without a remainder. That is, these numbers are divisible by 2. Numbers such as 2, 4, 6 and 8, which when divided by
2 have zero remainder, are called even numbers. 0 is also an even number.
When a whole number is divided by 2, if the remainder is zero, then the number is an even number.
Accordingly, even numbers starting from zero can be written as 0, 2, 4, 6, 8, 10, 12, ...
From the above information we see that when the number of pens is 3 or 5, then the pens cannot be divided equally between Nimali and Vimali. When an equal number of pens are given to both, then in the end, there is one remaining.
Numbers such as 1, 3, 5, 7, 9 and 11, which when divided by 2, have a non-zero remainder, are called odd numbers.
When a whole number is divided by 2, if the remainder is one, then the number is an odd number.
Accordingly, odd numbers starting from one can be written as 1, 3, 5, 7, 9, 11, 13, ...
Note
When the two even numbers 2 and 6 are added, the result is 8, which is also an even number. When we add any two even numbers in this manner, the result is also an even number.
We can verify the following statements by using the examples.
When two odd numbers are added together, the result is an even number.
When an even number and an odd number are added together, the result is an odd number.
When an even number is subtracted from an even number, the result is an even number.
When an odd number is subtracted from an odd number, the result is an even number.
When an odd number is subtracted from an even number, the result is an odd number.
When an even number is subtracted from an odd number, the result is an odd number.
When two odd numbers are multiplied together, the result is also an odd number.
When any whole number is multiplied by an even number, the result is an even number.
(2) From the numbers given below, choose the even numbers. Write them in the first row of the table below. Choose the odd numbers, and write them in the second row.
6" 7" 21" 24" 30" 35" 62" 70" 59" 100" 87" 71" 93" 94
(3) State whether the answer of each of following expressions is even or odd without solving.
●Identifying even numbers and odd numbers by considering the digit in the ones place.
Let us consider another method of finding out whether numbers like 2157 and 34 826 are odd numbers or even numbers without dividing the number by 2.
Now let us consider several whole numbers. Let us write each of them using the numbers its digits represent.
124 = 100 + 20 + 4
230 = 200 + 30 + 0
395 = 300 + 90 + 5
761 = 700 + 60 + 1
842 = 800 + 40 + 2
2157 = 2000 + 100 + 50 + 7
34 826 = 30 000 + 4000 + 800 + 20 + 6
The numbers represented by the digits in the tens place, hundreds place and the thousands place of any whole number given above, are multiples of 10. Therefore, those numbers are divisible by 2 (without a remainder). So, if the digit in the ones place is divisible by 2, (with zero remainder) then the given number is divisible by 2.
If the ones place of a number has one of the digits 0, 2, 4, 6 or 8 then the number is an even number.
If the ones place of a number has one of the digits 1, 3, 5, 7 or 9 then the number is an odd number.
(1) Write the even numbers between 10 and 25.
(2) Write the odd numbers from 19 till 35.
(3) Write the even numbers from 13 to 24.
(4) From the list of numbers given below, write the even numbers and the odd numbers separately.
456, 395, 714, 1852, 341, 27 850, 148 400, 397 659, 8 000 008
(5) Write your year of birth, month of birth and the date of birth respectively. Write whether each of them is an odd number or an even number.
(6) In a certain street of a city, the following sign is posted, “Vehicles can be parked here only on odd days of the month”. What are the days in the month of June that vehicles can be parked along the street?
(7) Write 5 even numbers and 5 odd numbers that can be written using the digits 4, 2, 3, 1 and 0 exactly once.
Recall what you learnt in the Factors lesson. Now let us find the factors of several numbers.
Each of the numbers 2, 3, 5 and 7 has only two different positive factors.
Each of the numbers 4, 6, 8 and 9 has more than two different factors.
Whole numbers greater than one such as 2, 3, 5 and 7 which have exactly two distinct factors are called prime numbers.
Let us now write the prime numbers from 1 to 20.
They are 2, 3, 5, 7, 11, 13, 17 and 19.
Of these prime numbers only 2 is even. All the others are odd.
Apart from 2, all the other even numbers have more than two factors.
Therefore, of all prime numbers, 2 is the only even prime number.
Whole numbers such as 4, 6, 8 and 9 which have more than two distinct
factors are called composite numbers.
Therefore whole numbers greater than one, other than prime numbers
are called composite numbers.
1 is neither a prime nor a composite number.
(1) The month of January of a certain calendar is given below. Draw
circles around the prime numbers and draw triangles around the
composite numbers.
(2) Copy the figure given below. From the given numbers, choose the
prime numbers. Draw an arrow from each prime number to the circle
indicating “prime numbers”. From the given numbers, choose the
composite numbers. Draw an arrow from each composite number to
the circle indicating “composite numbers”.
(3) (i) Write two successive prime numbers.
(ii) Write two successive composite numbers.
(4) Find the odd number between 1 and 10 which is not a prime.
(5) (i) Find a pair of prime numbers whose addition is 30.
(ii) Write 14 as a product of two prime numbers.
(6) (i) What is the smallest prime number?
(ii) What is the smallest composite number?
(7) Write the composite numbers between 20 and 30.
(8) What is the only even prime number?
Several numbers that can be represented using dots, arranged in a square formation are given below.
In each of the given boxes, the number of dots in a row is equal to the
number of dots in a column. By multiplying these two numbers the
number represented by the dots can be obtained.
That is,
1 = 1 × 1
4 = 2 × 2
9 = 3 × 3
16 = 4 × 4
We can write 25, 36 and 49 also in the above manner. Such numbers are
called square numbers.
(1) Look at the dates in the month of January in a calendar. Which dates
are square numbers?
(2) Write down the square numbers from 1 to 100.
(3) Write down the square numbers between 50 and 150.
(4) Add the odd numbers between 0 and 6. Find whether the sum is a
square number.
(5) Add the odd numbers between 0 and 10. Find whether the sum is a
square number.
In a certain shop, pipes are arranged as in the figure below.
The manner in which the pipes are seen from the front is,
It takes the shape of a triangle. Let us find out the number of pipes here.
The number of pipes in each row from the top is 1, 2, 3, 4, 5 and 6.
By adding these numbers, the total number of pipes is obtained as 21.
Therefore 21 can be represented using dots, where the dots are placed
in a triangular formation.
Let us find out about such numbers that can be represented using dots,
arranged in a triangular formation.
Numbers that can be represented as above are called triangular
numbers.
Considering the number of dots in each row,
Triangular number represented in the first figure = 1 = 1
Triangular number represented in the second figure = 1 + 2 = 3
Triangular number represented in the third figure = 1 + 2 + 3 = 6
Triangular number represented in the fourth figure = 1 + 2 + 3 + 4 = 10
Triangular number represented in the fifth figure = 1 + 2 + 3 +4 +5 = 15
As explained above, by starting at 1 and adding successive numbers to
it, we can obtain triangular numbers. Therefore, if we want to obtain
the tenth triangular number, we need to start at 1 and add to it the
successive numbers, up to 10.
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55
That is, the 10th triangular number is 55.
(3) What is the smallest triangular number?
(4) A number is represented using dots that are arranged in a triangular
formation. Then the bottom row consists of 11 dots. What is the
triangular number?
(5) Add two consecutive numbers given in exercise (2). Is the sum a
square number?
Let us write the even numbers in ascending order, starting at 2.
2" 4" 6" 8" 10" '''
This is the even number pattern starting at 2 written in ascending order.
Let us write the odd numbers in ascending order, starting at 1.
1" 3" 5" 7" 9" '''
This is the odd number pattern starting at 1 written in ascending order.
Let us write the square numbers in ascending order, starting at 9.
9" 16" 25" 36" '''
This is the square number pattern starting at 9 written in ascending
order.
1, 3, 6, 10, 15 is the triangular number pattern written in ascending
order.
Each number in a number pattern which has been arranged as above
according to a mathematical rule is called a term in that pattern.
(1) 1" 3" 6" 10" ''' is the triangular number pattern starting at 1, written in
ascending order. Find the 8th term.
(2) (i) 1, 4, 9, 16, ... is the square number pattern starting at 1, written
in ascending order. Find the 12th term.
(ii) In the above square number pattern, which term is 49?
(iii) Is 65 a term in this number pattern?
(iv) What are the terms that are between 50 and 100 in this pattern?
(3) Write the first 5 numbers in the number patterns given below.
(i) The pattern consisting of all even numbers greater than 5, written
in ascending order.
(ii) The pattern consisting of all multiples of 3 greater than 10,
written in ascending order.
(iii) The pattern starting at 1 and consisting of triangular numbers
that are not primes, written in ascending order.
(1) 1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32
(i) As given in the above order, write the numbers from 1 to 50
in your exercise book.
(ii) Now draw a line across 1.
(iii) Draw a circle around 2.
(iv) Now draw lines across all multiples of 2 other than 2.
(v) Draw a circle around 3.
(vi) Now draw lines across all multiples of 3 other than 3.
(vii) Draw a circle around 5. Now draw lines across all multiples
of 5 other than 5.
(viii) Draw a circle around 7. Now draw lines across all multiples
of 7 other than 7.
(ix) Now draw circles around the remaining numbers. Confirm
whether the numbers inside the circles are prime numbers?
(2) Amali says that, when we consider two consecutive whole numbers,
one of them is an even number while the other is an odd number. Is
this statement true or false?
(3) Jayamini explains that, by adding two consecutive triangular
numbers, we obtain a square number, by using the examples below.
1 + 3 = 4 3 + 6 = 9
Check whether her statement is true. Give 3 more examples to
verify this.
(4) Consider the sentences below. Mark the ones that are correct as
“” and the others as “”.
(i) 1 is a prime number.
(ii) The smallest prime number is 2.
(iii) All square numbers are composite numbers.
(iv) All triangular numbers are composite numbers.
(v) 36 is a composite number which is a square number as well as
a triangular number.
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