**Formulas:-**

0, 1, 2, 3, ----9. are called digits.

10, 11, 12,----- are called Number.

**Natural number (N) :-**

Counting numbers are called natural numbers.

Example:- 1, 2, 3,---etc. are all natural numbers. minimum natural number 1 and maximum natural number ∞

**Whole numbers (W) :-**

All counting numbers together with zero from the set of whole numbers

Example:- 0, 1, 2, 3, 4, ------ are whole number.

**Integers (Z) :-**

All counting numbers, 0 and -ve of counting numbers are called integers.

Example:- -∞---------, -3, -2, -1, 0, 1, 2, 3, -------∞

**Rational Numbers (Q) :-**

A Rational Number is a real number that can be written as a simple fraction

Example:- {p/q/p,q∈Z}

**Irrational NUmbers :-**

An Irrational Number is a real number that cannot be written as a simple fraction.

Example:- √

**Even numbers :-**

A number divisible by 2 is called an even number.

Example:- 0, 2, 4, 6, - - - - - - - - -

**Odd numbers :-**

A number not divisible by 2 is called an odd number.

Example:- 1, 3, 5, 7, - - - - - -

**Composite Numbers :-**

Numbers greater than 1 which are not prime, are called composite numbers.

Example:- 4, 6, 8, 9, 10, - - - -. 6 -> 1,2,3,6.

**Prime Numbers:-**

A number greater than 1 having exactly two factors, namely 1 and itself is called a prime number. Upto 100 prime numbers are:

**2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 71, 73, 79, 83, 89, 97**

**Co-prime Numbers:-**

Two natural numbers a and b are said to be co-prime if their HCF is 1.

Example:- (21, 44), (4, 9), (2, 3), - - - - -

**Twin prime numbers :-**

A pair of prime numbers (as 3 and 5 or 11 and 13) differing by two are

called twin prime number.

Example:- The twin pair primes between 1 and 100 are

**(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), (59, 61), (71, 73).**

**Face value:-**

Face value is the actual value of the digit.

Example:- In the number 7635, the "7" has a face value of 7, the face value of 3 is 3 and so on

**Place value:-**

The value of where the digit is in the number, such as units, tens, hundreds, etc.

Example:- In 352, the place value of the 5 is "tens"

Place value of 2 * 1 = 2;

Place value of 5 * 10 = 50;

Place value of 3 * 100 = 300.

1.

What is the place value of 7 in the numeral 2734?

**Answer: Option 'C'**

**7 × 100 = 700**

2.

What is the place value of 3 in the numeral 3259

**Answer: Option 'D'**

**3 × 1000 = 3000**

3.

What is the diffference between the place value of 2 in the numeral 7229?

**Answer: Option 'C'**

**200 - 20 = 180**

4.

What is the place value of 0 in the numeral 2074?

**Answer: Option 'D'**

**Note : The place value of zero (0) is always 0. It may hold any place in a number,
its value is always 0.**

5.

What is the diffference between the place value and face value of 3 in the numeral 1375?

**Answer: Option 'C'**

**place value of 3 = 3 × 100 = 300
face value of 3 = 3
300 - 3 = 297**

6.

A number when divided by a divisor leaves a remainder of 24.

When twice the original number is divided by the same divisor, the remainder is 11. What is the value of the divisor?

**Answer: Option 'B'**

**Let the original number be 'a'
Let the divisor be 'd'
Let the quotient of the division of aa by dd be 'x'
Therefore, we can write the relation as a/d = x and the remainder is 24.
i.e., a=dx+24 When twice the original number is divided by d, 2a is divided by d.
We know that a=dx+24. Therefore, 2a = 2dx + 48
The problem states that (2dx+48)/d leaves a remainder of 11.
2dx2dx is perfectly divisible by d and will therefore, not leave a remainder.
The remainder of 11 was obtained by dividing 48 by d.
When 48 is divided by 37, the remainder that one will obtain is 11.
Hence, the divisor is 37.**

7.

The largest number amongst the following that will perfectly divide 101^{100} – 1 is:

**Answer: Option 'C'**

**
The easiest way to solve such problems for objective exam purposes is trial and error or by back
substituting answers in the choices given.
101**

8.

In an election, candidate A got 75% of the total valid votes. If 15% of the total votes were declared invalid and the total numbers of votes is 560000, find the number of valid vote polled in favour of candidate.

**Answer: Option 'D'**

**Total number of invalid votes = 15 % of 560000
= 15/100 × 560000
= 8400000/100
= 84000
Total number of valid votes 560000 – 84000 = 476000
Percentage of votes polled in favour of candidate A = 75 %
Therefore, the number of valid votes polled in favour of candidate A = 75 % of 476000
= 75/100 × 476000
= 35700000/100
= 357000**

9.

Aravind had $ 2100 left after spending 30 % of the money he took for shopping. How much money did he take along with him?

**Answer: Option 'C'**

**Let the money he took for shopping be m.
Money he spent = 30 % of m
= 30/100 × m
= 3/10 m
Money left with him = m – 3/10 m = (10m – 3m)/10 = 7m/10
But money left with him = $ 2100
Therefore 7m/10 = $ 2100
m = $ 2100× 10/7
m = $ 21000/7
m = $ 3000
Therefore, the money he took for shopping is $ 3000.**

10.

A shopkeeper bought 600 oranges and 400 bananas. He found 15% of oranges and 8% of bananas were rotten. Find the percentage of fruits in good condition.

**Answer: Option 'A'**

**Total number of fruits shopkeeper bought = 600 + 400 = 1000
Number of rotten oranges = 15% of 600
= 15/100 × 600 = 9000/100 = 90
Number of rotten bananas = 8% of 400
= 8/100 × 400 = 3200/100 =32
Therefore, total number of rotten fruits = 90 + 32 = 122
Therefore Number of fruits in good condition = 1000 - 122 = 878
Therefore Percentage of fruits in good condition = (878/1000 × 100)%
= (87800/1000)% = 87.8%**

**Division Algorithm:-**If we divide a number by another number, then

**Dividend = (Divisor * Quotient) + Remainder**

D = d * Q + R

Example :- 7 = 3 * 2 + 1

**Divisibility Rules**

**Divisibility by 2:-**

The last digit of the given number should be 'zero' or 'even'

Example:- 12, 2498760, 34562, 745974 is divisible by 2.

**Divisibility by 3:-**

The sum of the digits of the number should be divisible by 3.

Example:- 1.) 123 => 1 + 2 + 3 = 6 = 6/3 = 2

Example:- 2.) 72954 => 7 + 2 + 9 + 5 + 4 = 27/3 = 9

**Divisibility by 4:-**

The last two digit of the given number should be 'zero' or 'divisible by 4.

Example:- 1.) 65837348 is divisible by 4, last two digits 48 is divisible by 4

Example:- 2.) 09214900 is divisible by 4, last two digits 00.

**Divisibility by 5:-**

The last digit of the given number should be 'zero' or '5'.

Example:- 1.) 8421745 is divisible by 5,

Example:- 2.) 9497440 is divisible by 5.

**Divisibility by 6:-**

The given number should be divisible by both 2 and 3.

Example:- 1.) 47562 is divisible by 2 as well as 3. so it is divisible by 6.

Example:- 2.) 59676 is divisible by 2 as well as 3. so it is divisible by 6.

**Divisibility by 8:-**

The last three digit of the given number should be 'zero' or 'divisible by 8.

Example:- 1.) 5478924568 is divisible by 8, last three digits 568 is divisible by 8

Example:- 2.) 69214000 is divisible by 8, last two digits 000.

**Divisibility by 9:-**

The sum of the digits of the number should be divisible by 9.

Example:- 1.) 786546 => 7 + 8 + 6 + 5 + 4 + 6 = 36 = 36/9 = 4

Example:- 2.) 72 => 7 + 2 = 9/9 = 1

**Divisibility by 11:-**

A number is divisible by 11, if the difference of the sum of its digits at odd places

and the sum of its digits at even places is either 0 or a nmber divisible by 11.

Example:- 1331, 14641, 123123, 9141.

11.

On dividinng 109 by a number, the quotient is 9 and the remainder is 1. Find the divisor.

**Answer: Option 'B'**

**d = (D-R)/Q
= (109 - 1)/9
= 108/9 = 12**

12.

What is the dividend. divisor 17, the quotient is 9 and the remainder is 5.

**Answer: Option 'C'**

**D = d × Q + R
D = 17 × 9 + 5
= 153 + 5
D = 158**

13.

In a division sum, the divisor is ten times the quotient and five times the remainder. If the remainder is 46, the dividend is:

**Answer: Option ''**

**Divisor = (5 × 46) = 230
= 10 × Quotient = Divisor
=> Quotient = 230/10 = 23
Dividend = (Divisor × Quotient) + Remainder
Dividend = (230 × 23) + 46 = 5336**

14.

In a division sum, the remainder is 6 and the divisor is 5 times the quotient and is obtained by adding 2 to the thrice of the remainder. The dividend is:

**Answer: Option 'C'**

**Divisor = (6 × 3) + 2 = 20
5 × Quotient = 20
Quotient = 4.
Dividend = (Divisor × Quotient) + Remainder
Dividend = (20 × 4) + 6 = 86**

15.

In a question on division with zero remainder, a candidate took 12 as divisor instead of 21. The quotient obtained by him was 35. The correct quotient is:

**Answer: Option 'D'**

**Number = (35 × 12) = 420
Correct quotient = 420/21 = 20**

16.

How many numbers from 10 to 50 are exactly divisible by 3.

**Answer: Option 'B'**

**12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,48.
13 Numbers.
10/3 = 3 and 50/3 = 16 ==> 16 - 3 = 13. Therefore 13 digits.**

17.

How many numbers from 10 to 100 are exactly divisible by 9.

**Answer: Option 'C'**

**10/9 = 1 and 100/9 = 11 ==> 11 - 1 = 10. Therefore 10 digits.**

18.

How many numbers from 29 to 79 are exactly divisible by 11.

**Answer: Option 'A'**

**29/11 = 2 and 79/11 = 7 ==> 7 - 2 = 5 Numbers**

19.

How many numbers from 2 to 7 are exactly divisible by 2.

**Answer: Option 'B'**

**3 - 1 = 2
2 + 1 = 3 Numbers.**

20.

How many numbers from 10 to 100 are exactly divisible by 10.

**Answer: Option 'D'**

**10/10 = 1 and 100/10 = 10
==> 10 - 1 = 9
==> 9 + 1 = 10 Numbers**

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