# Basic Concepts Of Whole Numbers

Whole Numbers are a part of the Number System and are an important topic for various competitive exams like SSC, Banking, and UPSC CDS. It covers 1-2 questions in the SSC tier 1 exam and covers 8-9 questions in main exams. This topic is an easily understandable and scoring topic.

__Whole Numbers:-__

Whole numbers are the numbers that are a set of Natural numbers along with zero. A set of whole numbers is given as {0,1,2,3,4……..}. It is denoted by W. W = {0,1,2,3…..}

__Some Facts About Whole Number:-__

a) All positive numbers are whole numbers, including 0.

b) All whole numbers are real numbers.

c) All the Natural numbers are whole numbers. Natural Numbers start with 1 except 0.

d) 0 is the smallest whole number.

e) Fractions, decimals, and negative numbers are not considered whole numbers as well as natural numbers.

0 is the number which is neither positive nor negative.

__Comparison between Whole Numbers and Natural Numbers:-__** **

From the above definition, it is clear that all-natural Numbers are whole numbers and all whole numbers are natural numbers except 0.

Set of Natural numbers:- {1,2,3,4……}

__Properties of Whole Numbers:-__

**Addition Property:- **If 0 is added to a whole number, then the result is the number itself. For Example:- 7+0 = 7.

**Multiplication Property:- **If 1 is multiplied by a whole number, then the result is the number itself. For example:- 7×1 = 7. If the whole number is multiplied by 0 then the result is 0. For example:- 7×0 = 0.

**Division Property:- **If a whole number is divided by 0 then the result is not defined. For example:- 7/0 = not defined.

**Distributive Property:- **This property is represented as P×(Q+R) = (P×Q)+(P×R). It is applicable for both addition and subtraction. For example:- let P=11, Q=12, R=14, 11×(12+14) = (11×12)+(11×14) = 286

**Commutative Property:- **This property is represented as P+Q = Q+P. This property is also applicable for multiplication, but not for subtraction and division. For example:- P=11, Q=12, 11+12 = 12+11 = 23.

__Questions on Whole Numbers:__

**Q1. A number in which one-sixth part is increased by 30 is equal to one-eighth part is decreased by 100. Find the number.**

a) 225

b) 240

c) 196

d) 216

**Solution:- **Let the number be x.

x/6 + 30 = 100 – x/8

x/6 + x/8 = 70

7x/24 = 70

x = 240

The number is 240. Option b) is correct.

**Q2. The Product of two numbers is 140 and the sum of squares of numbers is 449. Find the sum of both numbers.**

a) 24

b) 72

c) 33

d) 27

**Solution:- **let the two numbers be P and Q respectively.

P×Q = 140 ……..(1)

P²+Q² = 449 …..(2)

(P+Q)² = P²+Q²+2PQ

(P+Q)² = 449+2×140 = 449+280 = 729

P+Q = 27

Hence, option d) is the correct.

**Q3. Which is the largest five-digit number divisible by 91?**

a) 99190

b) 99008

c) 99099

d) None of these

**Solution:- **The Largest 5-digit number:- 99999

Largest five-digit number divisible by 91:- 91×1090 = 99190

**Q4. If the number 35x47x is divisible by 6, then what will be the value of x?**

a) 4

b) 5

c) 6

d) 7

**Solution:- **Divisibility of 6:- The number should have to be divisible by 2 & 3 both.

**Divisibility of 3:-** sum of numbers divisible by 3 then number divisible by 3

Possible values of x:- 1, 4,7

1 and 7 are not possible because both values can’t divisible by 2. So the value of x is 4.

Hence, option a) is correct.

**Q5. Which of the following numbers is divisible by 13?**

a) 8970

b) 8465

c) 7814

d) 9765

**Solution:- **Multiply the unit digit of the number by 9 and subtract it from the rest of the number. If the resultant number is divisible by 13 then the number is divisible by 13.

Unit digit 0 × 9 = 0

897 – 0 = 897

8970 is divisible by 13. Hence, option a) is correct.

**Divisibility rule of 13:- **Multiply unit digit with 9 and subtract from the remaining number.

**Q6. 3⁵⁵ + 3⁵⁶ + 3⁵⁷ + 3⁵⁸ is divisible by which of the following. **

a) 2

b) 5

c) 4

d) All of these

**Solution:- **3⁵⁵(1+3+3²+3³)

3⁵⁵ × 40 which is divisible by 2, 3,4,5,10

i.e 3×40 = 120 is divisible by 2,3,4,5,6,8,10,12 etc. So the number is divisible by all the given options.

Hence, option d) is correct.

**Q7. Find the value of 1³+2³+3³+…….+11³. **

a) 4356

b) 4576

c) 4646

d) 4816

**Solution:- **Sum of cubes of natural numbers:- [n(n+1)/2]²

⇒ n = 11

⇒ [11(11+1)/2]² = 121×36

⇒ 4356

Hence, option a) is correct.

**Q8. What is the value of 101, 104, 107…….131? **

a) 1466

b) 1576

c) 1276

d) 1386

**Solution:- **All numbers are in Arithmetic Progression

Common difference d = 104-101 = 3

Final term a_{n }= a + (n-1)d

Where a is the first term, n is number of terms

131 = 101 + (n-1)3

30 = 3n-3

n = 11

Sum of numbers in A.P = S_{n }= n/2(a+l) where l = last term

S_{n }= 11/2 (101+131) = 1276

Hence, option c) is correct.

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