** Teachers Eligibility Test - Paper 2 **

**Class – VI Mathematics**

**NUMBER SYSTEM**

**
**

**Odd and Even Numbers **

A
number is called an **odd
number**** **if it cannot be grouped

Equally in twos. 1, 3, 5, 7, …, 73, 75, …,
2009,… are

**Odd numbers.**

All odd numbers end with anyone of the digits 1, 3, 5, 7 or 9.

A number is called an **even number**** **if it can be
grouped

Equally in twos. 2, 4, 6, 8... 68, 70, . . ,
4592... Are

**Even numbers.**

All even numbers end with anyone of the digits
0, 2, 4, 6 or .

In
whole numbers, odd and even numbers come alternatively.

**Prime Numbers**

**
**A natural number
greater than 1, having only two factors namely 1 and the number itself, is
called a** ****prime
number.**

**For example**, 2 (1 x 2) is a prime number as is 13 (1 x 13).

**Composite Numbers**

**
**A natural number
having more than 2 factors is called a** ****composite number.**

**For example**, 15 is a composite number (15 = 1 x 3 x 5) as is 70 (1 x
2 x 5 x 7).

**Expressing
a Number as the Sum of Prime Numbers**

Any
number can always be expressed as the sum of two or more prime numbers.

**Example
1:**

Express 42 and 100 as the sum of two consecutive primes.

**Solution: **

** **42 = 19+23;

100 = 47+53

**Example
2: **

Express 31 and 55 as the sum of any three odd primes.

**Solution: **

** **** **31 = 5+7+19 (find another way, if possible!)

55 = 3 + 23+29

**Twin
Primes**

A pair of prime numbers, whose difference is
2, is called **twin primes**.

**For example**, (5, 7) is a twin prime pair as is (17, 19).

**Rules for Test of Divisibility of Numbers**

**Divisibility by 2**

**
**A number is **divisible by 2**, if its ones place is any one of the even numbers 0, 2, 4, 6 or 8.

**Examples:**

1. 456368 are divisible by 2, since its ones
place is even (8).

2. 1234567 are not divisible by 2, since its
ones place is not even (7).

**Divisibility by 3**

Divisibility of a number by 3
is interesting! We can find that 96 are divisible by 3.

Here, note that the sum of its digits 9+6 =
15 is also divisible by 3.

Even 1+5 = 6 is also divisible by 3. This is
called as **iterative** or **repeated **addition.

So, a number is **divisible by 3** if the sum of its digits is divisible by 3.

**Examples:**

1. 654321 are divisible by 3.

Here 6+5+4+3+2+1= 21 and 2+1=3 is divisible
by 3.

Hence, 654321 are divisible by 3.

2. The sum of any three consecutive numbers
is divisible by 3.

(For example: 33+34+35=102, is divisible by
3)

3. 107 is not divisible by 3 since 1+0+7=8,
is not divisible by 3.

**Divisibility by 4**

A number is **divisible
by 4** if the last two digits of the given number are **divisible by 4**.

Note that if the last two digits of a number
are zeros, then also it is **divisible by 4.**

**Examples: **

**
**71628, 492, 2900 are divisible by 4, because 28 and 92 are divisible by 4
and 2900 is also divisible by 4 as it has two zeros.

**Divisibility by 5**

Observe the multiples of 5.

They are 5, 10, 15, 20, 25,.., 95, 100, 105,
…., and keeps on going.

It is clear, that multiples of 5 end either
with 0 or 5 and so, A number is **divisible by 5** if its
ones place is either 0 or 5.

**Examples: **

5225 and 280 are divisible by 5

**Divisibility by 6**

A number is **divisible
by 6** if it is divisible by both 2 and 3.

**Examples**:

138,
3246, 6552 and 65784 are divisible by 6.

**Divisibility by 8**

A number is **divisible by 8** if the last three digits of the given number are **divisible by 8**.

Note that if the last three digits of a
number are zeros, then also it is **divisible by 8.**

**Examples:**** **

2992 is divisible by 8 as 992 is divisible by 8 and 3000 is
divisible by 8 as its last three digits are zero.

**Divisibility by 9**

**
**A** **number is **divisible by 9** if the sum of its digits is divisible by 9.** **Note that the numbers
divisible by 9 are divisible by 3.

**Examples: **

**
**9567 is divisible
by 9 as 9+5+6+7=27 is divisible by 9.

**Divisibility by 10**

**
**A number is **divisible by 10** if its ones place is only zero.** **

Observe that numbers divisible by 10 are also
divisible by 5.

**Examples:**

1. 2020 is divisible by 10 (2020÷10 = 202) whereas
2021 is not divisible by 10.

2. 26011950 is divisible by 10 and hence
divisible by 5.

**Divisibility by 11**

A number is **divisible
by 11** if the difference between the sum of
alternative digits of the number is **either 0 or divisible by 11**.

**Examples: **

Consider the number 256795.

Here, the difference between the sum of alternative digits =(2+6+9)−(5+7+5)=17−17=0.

Hence, 256795 are divisible by 11.

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