Number system with solved examples

1.Natural numbers:

Counting Numbers (1,2,3,4………) are called Natural numbers

2.Whole number: Counting Numbers starting from zero. (0,1,2,3,4………………

0 is not a natural number.

Every natural number is a whole number.

3.Integers: Positive Integer or Non negative integer.0,1,2,3,4...So on

Negative integer or Non-positive integer   -1, -2, -3…. so on

4.Even numbers: Numbers divided by 2.

5.Odd numbers: Numbers not divisible by 2.

6.Prime numbers: Number divisible by 1 or itself only.

Eg. 2, 7,11,13,23 etc.

7.Composite numbers: Number that has more than 2 distinct factors.

Eg. 6,8, 14

1 is neither prime or composite number.

8. Co-prime numbers: Two natural numbers are said to be co-prime if their HCF is 1 as (2,3) (8,11)

Rule. To test whether a number is prime or not

Steps to test a prime number.

If the given number is P then.

Step 1. Find the whole number x such that x > P

Step 2.Take all prime numbers less than or equal to x.

Step 3.If none of these divides p exactly then p is a prime number otherwise it is non-prime.

Ex.1 P = 193

Sol. 14> √193

Prime numbers less than 14 are 2,3,5,7,11,13

None of these divides 193 exactly thus 193 is a prime number.

Ex 2. Is 881 a Prime number?

Sol. The Approx square Prime number less than 30 are 2,3,5,7,11,13,17,19,23,29 root of 881 is 30

is not divisible by and of the above number, so it is a prime number.

Test of Divisibility:

Divisibility by 2: A number is divisible by 2 if it’s a unit digit is 0,2,4,6

Divisibility by 3: A number is divisible by 3 if the sum of the digit is divisible by 3.

Ex. 2553.

Sol. 2+5+5+3 = 15

Which is divisible by 3.

Divisibility by 4: A number is divisible by 4 if the last two digits are divisible by 4.

Divisibility by 5: A Number is divisible by 5 if it’s a unit digit is 5 or 0.

Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3 or u can say the number is even and divisible by 3

Ex. 4536 is an even number.

Sol.4+5+3+6 = 18 divisible by 3.

Therefore, it is divisible by 6

Divisibility by 7: The unit digit is doubled then subtracted from the omitted number.

Ex. Check if 133 is divisible by 7.

Sol. 13-3*2 =13-6= 7 which is divisible by 7

Check if 4606 is divisible by 7

Step 1. 460-6*2= 448

Step 2. 44-8*2=28 

which is divisible by 7

Divisibility by 8: If the last three digits of the number are divisible by 8.

Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.

Divisibility by 10: If the unit digit is 0 then the number is divisible by 10.

Divisibility by 11: If the difference of the sum of the digit at odd and even places and some of the digits at even places is either zero or divisible by 11.

Ex. find 1331 is divisible by 11?

Sol. Sum of the digit at odd place 3+1=4

Sum of the digit at even place 3+1=4

Their difference =4-4=0. Therefore, it is divisible by 11

Factors of composite numbers:

Composite numbers are the numbers that can be factorized into prime factors or we can say composite numbers are the numbers that are not prime.

Composite number = P₁^x1  * P₁^x2   *…… P_n^xn   

P₁, P₂, P_n  are prime numbers

X₁, X₂, X_n are the power factors

Factors of composite numbers = (x₁+1) (x₂+1).....(x_n+1)

Ex. Find factors of 18.

Sol. Factors of 18 = 2*3*3 = 21 * 32

Factors of 18 is (1+1 )*( 2+2) = 2*3= 6 ans

Fractions:

The word fraction means a part of anything. It can be expressed in the form p/q where p and q are integers and q is not equal to 0.

P is called the numerator and q is called the denominator.

Ascending or descending order or the fractions:

Rule 1.when the numerator and the denominator of the fractions increase by a constant value, the last fraction is the biggest.

Ex 1.Which one of the following fractions is the greatest 3/4,4/5,5,6?

Sol. we see that the numerators, as well as denominators of the above fractions, increase by 1, so the last fractions 5/6 is the greatest

Ex2. Which one of the following fractions is the greatest 1/8,4/9,7/10?

Sol. We see that the Numerator increases by 3 and the denominator also increases by a constant value of 1,

so the last fraction 7/10 is the greatest.

Rule 2. The fraction whose numerator after cross-multiplication gives the greater value is greater.

Ex.Which is greater than 5/8 or 9/14.

Sol. For quicker calculation no need to calculate decimal numbers or by equating the denominator.

Cross multiply the two fraction

Step 1. 5/8 = 9/14

5*14 and 8*9

70 and 72

Step 2. 72 is greater than and the numerator involved with the greater value is 9, the fraction 9/14 is the greater of two.

Recurring fractions:

i. Pure Recurring fraction: A decimal fraction in which all the digits after decimal points are repeated is called a pure recurring decimal.

ii. Mixed Recurring fractions: A decimal fraction in which some figures are not repeated while some of them are repeated is called a mixed recurring decimal. Eg.

Method to convert recurring Decimal fraction to fraction:

To convert a pure recurring decimal to fraction:

Rule. Write a repeated figure or only one in the numerator without a decimal point and take as many nines as is the number of repeating digits.

Ex. .41

Sol.    41/99.

To convert a mixed recurring fraction

Rule. Take the difference between the numerator formed by all the digits after the decimal point and that formed by the digits which are not repeated, In the denominator takes the number formed by as many nines as in the number of followed as many zeros of non-repeating digits.

Ex.1. Express .17̅

Sol. (17-1)/90 = 8/45

Ex. 2.Express 2.634̅

Sol. 2+ (634-63)/900 = 2+ 571/900 = 2371/900

Counting numbers of zeros:

In this type of problem, we have to count the numbers of zeros at the end of the number.

10! = 10*9*8*7*6*5*4*3*2*1

We have to count the number of fives because multiplication of 5 with even numbers will result in 0 in the end.

In 10! We have 2 fives thus the total number of zeros is 2.

Short cut :

Value will be n/5 +n/5²+n/5³………..

The integral value of the sum is the total number of zeros.

Ex. Find the total number of zeros in 10!

Sol.10/5+10/52 here integral value is 2. Here 10/5² is less than 1 so we do not count it.

Ex. Find the number of zeros in 100!

Sol 100/5+100/5²+100/5³

Integral value is 20+4 = 24 zeros.