**Learn Numbers System – (Number System aptitude tricks)**

**Numbers System** is one of the most important topics from the **Advance maths** section from which you can expect lots of questions in SSC, Railway, Banking and other State level exams. A student preparing for these examinations should also be aware of that when it comes to Number System, mugging up just formulas of it is going to be of no use and the detailed understanding of the concepts and proper practice can help you out with the Number System topic during the examination.

Numbers are broadly classified into two categories:

- Real Numbers
- Imaginary Numbers

Further, Real numbers are also classified into two categories:

- Rational Numbers
- Irrational Numbers

Rational Numbers are further classified as:

- Fractions
- Integers

We classify integers into three categories as:

- Negative Integers
- 0 (Neither negative nor positive)
- Positive Integers

__ __

**Natural Numbers****: **Numbers which are used for counting the objects are called natural numbers. They are denoted by N.

N = {1, 2, 3………………..}

All positive integers are natural numbers.

**Whole numbers:** When ‘zero’ is included in the natural numbers, they are known as whole numbers.

They are denoted by W.

W= {0, 1, 2, 3……………….}

**Integers:** All natural numbers, zero and negatives of natural numbers are called as integers. They are denoted by I.

I = {………………..,-3, -2, -1, 0, 1, 2, 3………………}

**Rational numbers****: **The numbers which can be expressed in the form of P/Q where P and Q are integers and q ≠ 0 are called rational numbers.

E.g. 1/7, 2, -3

**Irrational numbers****: **The numbers which cannot be written in the form of P/Q where P and Q are integers and q ≠ 0 are called irrational numbers. E.g. √3, √7, π

When these numbers are expressed in decimal form, they are neither terminating nor repeating.

**Real numbers****:** Real numbers include both rational as well as irrational numbers.

Positive or negative, large or small, whole numbers or decimal numbers are all real numbers.

E.g. 1, 13.79, -0.01, etc.

**Imaginary numbers****: **An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit ‘i’ which is defined by its properly .

**Prime number****: **A prime number is a natural number greater than 1 and is divisible only by 1 and itself.

E.g. 2, 3, 5, 7, 11, 13, 17, 19 ………….etc.

**Note: **2 is the only even prime number.

**Composite Numbers****: **A number, other than 1, which is not a prime number is called a composite number .

E.g. 4, 6, 8, 9, 10, 12, 14, 15 ……….etc.

**Note: **1 is neither a prime number nor a composite number.

There are 25 prime numbers between 1 and 100.

Every prime number except 2 & 3 can be expressed in the form of (6k+1) or (6k-1) where k is an integer.

If a number ‘N’ is not divisible by any prime number less than, then N is a prime number.

**Co-prime numbers****: **Two numbers are said to be co-prime if their HCF is 1.

E.g. (2,3), (3,4) , (5,7), (3,13) etc.

**Even numbers:** The number which is divisible by 2 is called an even number.

E.g. 2, 4, 6, 8…………….

**Odd numbers:** The number which is not divisible by 2 is called an odd number.

E.g. 3, 5, 7, 9…………..

**Consecutive numbers:** A series of numbers in which the succeeding number is greater then the preceding number by 1 is called a series of consecutive numbers.

**Divisibility Rules**

**Divisibility by 2:**Number whose last digit is either even or zero is divisible by 2.**Divisibility by 3:**If the sum of the digits of a number is divisible by 3, then the number is also divisible by 3.**Divisibility by 4:**If the last two digits of a Number is divisible by 4 or the number having two or more zeros at the end, then the number is divisible by 4.**Divisibility by 5:**If the last digit of a number is either 5 or 0, then the number is divisible by 5.**Divisibility by 6:**If a number is divisible by both 2 and 3, then the number is also divisible by 6.**Divisibility by 8:**If the last three digits of a number is divisible by 8 or the last three digits of a number are zeros, then the number is divisible by 8.**Divisibility by 9:**If the sum of all the digits of a number is divisible by 9, then the number is also divisible by 9.**Divisibility by 10:**The number which ends with zero is divisible by 10.**Divisibility by 11:**If the sums of digits at odd and even places are equal or differ by a number divisible by 11, then the number is also divisible by 11.**Divisibility by 12**: The number which is divisible by both 3 and 4 is also divisible by 12.

**How to calculate Number of factors, Product of factors and Sum of all factors?**

These are certain formulas related to factors of a number N, such that, N= A^{p }x B^{q }x C^{r}, where, A, B and C are prime factors of the number N and p, q and r are non-negative powers/ exponents.

Number of factors of N = **(p+1) (q+1) (r+1)**

Product of factors of N = **N ^{total No. of factors/2}**

Sum of all factors of “N” =

*To learn the Number System tricks through video lecture..*

Now let’s discuss some questions on how to solve **Number problems** easily:-

**Eg: The difference of two numbers is 1365. On dividing the larger number by the smaller, we get 6 as quotient and the 15 as remainder. What is the smaller number? **

**Sol: **Let the smaller number be *x*. Then larger number = (*x* + 1365).

⇒ *x* + 1365 = 6*x* + 15

⇒ 5*x* = 1350

⇒ *x* = 270

So, Smaller number = **270.**

** **

**Eg: ****If the number 517*324 is completely divisible by 3, then the smallest whole number in the place of * will be:**

**Sol: **Sum of digits = (5 + 1 + 7 + x + 3 + 2 + 4) = (22 + x), which must be divisible by 3.

**⇒**** x = 2.**

**Eg: ****How many 3-digit numbers are completely divisible 6?**

**Sol****: **3-digit numbers divisible by 6 are: 102, 108, 114,…, 996

This is an A.P. in which a = 102, d = 6 and l = 996

Let the number of terms be n. Then t_{n} = 996.

a + (n – 1)d = 996

102 + (n – 1)6 = 996

6(n – 1) = 894

(n – 1) = 149 ⇒ n = 150

So, Number of terms = **150.**

**Eg: What is the unit digit in 7 ^{105}?**

**Sol: **Unit digit in 7^{105} = Unit digit in [(7^{4})^{26} x 7]

But, unit digit in (7^{4})^{26} = 1

⇒ Unit digit in 7^{105} = (1 x 7) = **7**

**Eg: Find the number of factors of 98 and also find the sum and product of all factors.**

**Sol:** First write the number 98 into prime factorization

98 = 2 x 49 = 2x 7 x 7

98 = 2^{1} x 7^{2 }Here A = 2, B = 7, p= 1, q = 2

Number of factors of the number 98 = (p + 1) (q +1) = 2 x 3 = **6**

**Sum of all factors of 98 = **

= 3 x 57 = **171**

Product of all the factors of number **98** = (98)^{6/2} = (98)^{3} = **941192**

Please share your feedback and doubts in the comment box section regarding the **Number System notes** discussed above which will be beneficial for your upcoming SSC exams, banking exams and Railway exams.

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