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**Natural Numbers**: The numbers which are used to count the objects known as natural numbers.

N = {1, 2, 3, …}

**Whole Numbers**: In the natural numbers set if we include number 0 then the resulting set will be known as whole numbers.

W = {0, 1, 2, 3, ….}

**Integers:** The natural numbers along with zero and their negatives are known as integers.

I = {…., -3, -2, -, 0, 1, 2, 3, …}

- Positive Integers: The set of all the natural numbers is known as positive integers.

I^{+} = {1, 2, 3, …}

- Negative Integers: The set of all negative numbers are known as negative integers.

I^{–}= {-1, -2, -3,….}

**Rational Numbers: **The set of numbers which can be expressed in the form of p/q** **where q ≠ 0 and is known as rational numbers.

Q= { p/q : p and q are integers and q ≠ 0}

e.g. -3, 4, 6/4, etc, are rational numbers.

**Irrational Numbers**: Those numbers which cannot be expressed in the form of p/q, where q ≠ 0, are known as irrational numbers. For e.g. √2, √5, √7 etc. are irrational numbers.

**Real Numbers: **The set of those numbers which are either rational or irrational are known as real numbers.

**Even Numbers: **Those numbers which are divisible by 2 and are known as even numbers. For e.g. 4, 6, 8, 10, … are even numbers.

**Odd Numbers:** Those numbers which are not divisible by 2 and are known as odd numbers. For e.g. 3, 5, 9, 7, … are odd numbers.

**Prime Numbers: **A prime number is a number which has no factors besides itself and unity, i.e. it is divisible only by itself and 1 but not by any other number.

**Composite Numbers: **A composite number is one which has other factors including itself and unity.

- 1 is neither prime nor composite.
- A composite number may be even or odd.

Test for Prime Numbers: For numbers less than 100, it is not very difficult to determine whether it is a prime number or not.

For testing any number greater than 100 whether it is a prime number or not.

- We take the nearest integer larger than the approximate square root of that number, suppose it is x.
- We test the divisibility of the given number by every prime number less than x.
- If the number is not divisible by any one of them, then it is a prime number otherwise it is a composite number.

**Perfect Number: **If the sum of the divisors of a number N excluding N itself is equal to N, then N is called a perfect number.

**Decimal Number: **A collection of digits (0, 1, 2, 3,…., 9) after a point is called a decimal fraction.

Test of Divisibility:

- Divisible by 2 – If in a given number there 0 or even number at unit’s place then that number is divisible by 2.
- Divisible by 3 – If the sum of digits of a given number is divisible by 3 then that number is divisible by 3.
- Divisible by 4 – If a number formed by using the unit’s and ten’s place digits of a given number, is divisible by 4, then that given number is divisible by 4.
- Divisible by 5 – If in a given number there is 0 or 5 at unit’s place then that number is divisible by 5.
- Divisible by 6 – If any given number is divisible by 2 and 3 then that given number is divisible by 6.
- Divisible by 8 – If a number formed by using last three digits of a given number is divisible by 8 then that given number is divisible by 8.
- Divisible by 9 – If the sum of digits of a number is divisible by 9 then that number is divisible by 9.
- Divisible by 11 – If the difference of the sum of digits in odd places and the sum of its digits in even places is either 0 or a multiple of 11 then that given number is divisible by 11.

**To find the unit’s place digit in the product of numbers**: To find the unit’s place digit in the product of number, we take units place digits of each number. Now, find the product of these digits. If there is any number (more than 9) in the product then we proceeds as above i.e. we again take unit’s place digits and find the product.

For e.g. units place digit in

249 * 349 * 577 * 344

= unit’s place digit in 9 * 9 * 7 * 4

= unit’s place digit in 81 * 28

= unit’s place digit in 1 * 8

= 8

**To find the Units Place Digit in a number of the form N ^{n}**

- If unit’s place digit of number is 0, 1, 5 or 6 then the units place digit in N
^{n}remains unaltered e.g. unit’s place digit in (576)^{1151}= 6. - If unit’s place digit of a number is 2, then divide the index of number by 4 then write in form of 2
^{4}and solve and unit’s place digit in 2^{4}= 6, e.g. unit’s place digit in (572)^{443}

= unit’s place digit in (2^{4})^{110} * 2^{3}

= unit’s place digit in 6 * 8

= unit’s place digit in 48 = 8

∴ Unit’s place digit in (572)^{443} = 8

- Similarly, we find the unit’s place digit of a given number which has unit’s place digit either 4 or 8.
- If there is unit’s place digit either 3 or 7 in given number then we solve the number as above but unit’s place digit in 3
^{4}= unit’s place digit in 7^{4}= 1. - If number contains 9 as unit’s place digit and index is odd, then the required unit’s place digit in N
^{n}is 9 and if index is even, then the required unit’s place digit in N^{n}is 1 e.g. unit’s place digit in (539)^{140}= 1 and unit’s place digit in (539)^{141}= 9

**Aptitude Questions (Number Systems):**

1. The difference between the place value and the face value of 5 in the numeral 426539 is

A) 363

B) 691

C) 495

D) 400

E) None of these

View AnswerAnswer: option (C)

Explanation:

Place value of 5 = 500

Face value of 5 = 5

Difference = Place value – Face value

= 500 – 5 = 495

2. The difference between the place value of two 7 in the number 96753271 is

A) 0

B) 6993

C) 69930

D) 699930

E) None of these

View AnswerAnswer: option (D)

Explanation:

Place value of first 7 = 700000

Place value of second 7 = 70

Difference = Place value of first 7 – Place value of second 7

= 700000 – 70 = 699930

3. The unit digit in the product (783 * 516 * 912 * 467) is:

A) 6

B) 2

C) 5

D) 3

E) None of these

View AnswerAnswer: option (B)

Explanation:

Unit digit in the given product = Unit digit in (3 * 6 * 2 * 7)

= 18 * 14 = 252

Unit digit in the given product is 2.

4. What is the unit digit in 8^{105}?

A) 1

B) 5

C) 8

D) 9

E) None of these

View AnswerAnswer: option (C)

Explanation:

Unit digit in 8^{105} = Unit digit in [(8^{4})^{26} * 8]

But, unit digit in (8^{4})^{26} = 6

∴ Unit digit in 8^{105} = 6 * 8 = 48 = 8

5. What is the unit digit in (6^{95} – 2^{58})?

A) 0

B) 2

C) 6

D) 7

E) None of these

View AnswerAnswer: option (B)

Explanation:

Unit digit in 6^{95} = Unit digit in [(6^{4})^{23} * 6^{3}] = 6 * 6 = 36 = 6

Unit digit in 2^{58} = Unit digit in [(2^{4})^{14} * 2^{2}] = 6 * 4 = 24 = 4

Unit digit in (6^{95} – 2^{58}) = (6 – 4) = 2