3. Introduction to Numbers
Type of Number Description
Counting Numbers {1, 2, 3, ...}
Whole Numbers {0, 1, 2, 3, ...}
Integers {..., -3, -2, -1, 0, 1, 2, 3, ...}
Rational Numbers p/q : p and q are integers, q is not zero
Irrational Numbers Not Rational
Real Numbers Rational Numbers and Irrational Numbers
Summary of Numbers
5. Divisibility Test
DIVISIBILITY BY 2
A number is divisible by 2, if its units digit is an even digit i.e. 0, 2, 4, 6, or 8.
DIVISIBILITY BY 3
A number is divisible by 3, if the sum of its digits is divisible by 3.
DIVISIBILITY BY 4
A natural number is divisible by 4, if the number formed by its digits in units and tens places is
divisible by 4.
Also, if the number formed by its digits in units and tens places is not divisible by 4, then the
given number is not divisible by 4.
Consider the number 79812. Since 12 is a multiple of 4. So, 79812 is divisible by 4.
Consider now the number n = 23472392. Since, 92 is a multiple of 4. So, n is divisible by 4. The
number m = 23798531 is not divisible by 4, because 31 is not a multiple of 4.
DIVISIBILITY BY 5
A number is divisible by 5 , if its unit digit is 0 or 5.
DIVISIBILITY BY 6
A number is divisible by 6 , if it is divisible by 2 as well as 3.
6. Divisibility Test
DIVISIBILITY BY 8
A natural number is divisible by 8, if the number formed by its digits in units, tens and hundreds
places is divisible by 8.
Consider the number 79816. Since 816 is a multiple of 8. So, 79816 is divisible by 8.
DIVISIBILITY BY 9
If the sum of the digits of a natural number is divisible by 9, then the number is divisible by 9.
ILLUSTRATION: If 21y5 is a multiple of 9, where y is a digit , what is the value of y?
SOLUTION: Since 21y5 is a multiple of 9. Therefore, the sum of its digits is a multiple of 9.
i. e. 2 + 1 + y + 5 is a multiple of 9
⇒ y + 8 is a multiple of 9
⇒ y + 8 = 0, or 9, or 18, or 27, 36, ... ...(i)
But, y is a digit. So, y can take values 0,1,2, …, 9.
.⋅. y + 8 can take values 8, 9, 10, 11, ..., 17. ...(ii)
From (i) and (ii), we get
y + 8 = 9 ⇒ y = 9 − 8 = 1
Hence,y = 1.
DIVISIBILITY BY 11
A number is divisible by 11, if the difference of its digits in odd
Places and the sum of its digits in even places is either 0 or a multiple of 11.
Consider three‐digit number 264. For this number, we have
Sum of the digits in odd places−Sum of the digits in even places = (2 + 4) − 6 = 0
So, 264 is divisible by 11.
Ex: 61809
7. Divisibility Test
Example: If 31z5 is a multiple of 3, where z is a digit, what might be the value of z?
Example: Without actual division find the remainder when 379843 is divided by 3.
8. Divisibility Test
Example: If 21y8 is a multiple of 6, where y is a digit, what might be the value of y?
Example: If 21y8 is a multiple of 6, where y is a digit, what might be the value of y?
9. Divisibility Test
Example: If 31z is a multiple of 11, where z is a digit, what is the value of z?
Example: Given that the number 148101a095 is divisible by 11, where a is some digit,
what are the possible values of a?
10. Divisibility Test
Example: Given that the number 1735538a05 is divisible by 9, where ′a′ is a digit, what are
the possible values of a?
Example: Given that the number 7713a8 is divisible by 4, where a is a digit, what are the
possible values of a?
12. HCF and LCM
Finding HCF
Method: (Short cut method) – Most popular method
2 |12, 16 (Divide by common divisor 2)
2 | 6, 8(Divide by common divisor 2)
3, 4
HCF = 2 x 2 = 4
Method: Continued Division Method
Divide the bigger number by smaller number. Then the divisors are divided in succession
by the remainders got. This division should be carried out till we get the remainder zero.
The last divisor is the HCF of the given numbers.
15. HCF and LCM
Example: Find the HCF and LCM of 12, 20, 32.
Example: The HCF of 72 and 252 is 36. Find their LCM.
16. HCF and LCM
Example: The students of a class can be divided into groups of 6 or groups of 8 without
leaving out any student. What will be the minimum number of students in such a class?
Example: A merchant has 120 liters and 180 liters of two kinds of oil. He wants to sell the
oil by filling the two kinds of oil in tins of equal volumes. What is the greatest volume of
such a tin?
