**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**

**Formulae:-**

The sum of first n natural numbers is =**[n(n+1)]/2**.

Example:- 1.) The sum first 10 natural number is =**[10(10+1)]/2 = [10(11)]/2 = 55**

Example:- 2.) The sum first 49 natural number is =**[49(49+1)]/2 = [49(50)]/2 = 49 * 25 = 1225**

The sum of the squares of the first n natural numbers is =**[n(n+1)(2n+1)]/6.**

Example:- 1.) The sum of the squares of the first 10 natural numbers is =**10(11)(21)/6 = 385.**

The sum of the cubes of the first n natural numbers is =**[n(n+1)/2]**^{2}

Example:- 1.) The sum of the cubes of the first 10 natural numbers is =**[10(10+1)/2]**^{2}

= [(10*11)/2]^{2}= 55^{2}= 3025

The sum of the first n odd natural numbers is =**n**^{2}

Example:- 1.) The sum of the first 24 odd natural numbers is =**24**^{2}= 576.

Example:- 2.) The sum of the first 9 odd natural numbers is =**9**^{2}= 81.

The sum of the first n even natural numbers is =**n(n+1)**

Example:- 1.) The sum of the first 10 even natural numbers is =**10(10 + 1) = 10 * 11 = 110**

Example:- 2.) The sum of the first 12 even natural numbers is =**12(12 + 1) = 12 * 13 = 156**

- 21. How many numbers from 22 to 90 are exactly divisible by 11.

**Answer: Option 'B'**

**22/11 = 2 and 90/11 = 8
==> 8 - 2 = 6
==> 6 + 1 = 7 Numbers**

- 22. If a worker in a factory receives one rupee on the first day from the second days onwards his wage is increased by one rupee every day. What is the total amount of wage he receives after 40 days?

**Answer: Option 'B'**

**sum of n natural number = n(n+1)/2
= 40(41)/2
= Rs.820/-**

- 23. A bell in a clock rings once at 1 O'clock, twice at 2 O'clock, thrice at 3 O'clock and so on.. Then how many times it rings in a day.

**Answer: Option 'D'**

**sum of n natural number = n(n+1)/2
= 12(13)/2 = 78
= 2 times (78) = 156**

- 24. What is the sum of natural numbers between 20 and 100.

**Answer: Option 'A'**

**a = first number
l = last number
S**

- 25. What is the sum of all two digit numbers?

**Answer: Option 'C'**

**10 + 11 + 12 + 13 + 14 + 15 + - - - - - - - - + 97 + 98 + 99 + 100
S _{n} = n/2[a + l]
= 90/2[10 + 99]
= 45[109]
= 4905.**

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