## Numbers Questions - 2

What is the remainder when 33125 is divided by 8 ?
• 1
• 2
• 3
• 4
• None
Explanation

Up to 32, 8 can divide exactly, we can express the number as (32 +1)125

As up to 32 exactly divisible by 8, and 1125 = 1, hence the remainder will be 1.

Workspace
A number when increased by $\displaystyle \frac{1}{4}$ results in a value 120. What is the number ?
• 96
• 100
• 80
• 90
• None
Explanation

Let the number be x, if it exceeds by $\displaystyle \frac{1}{4}$, the value becomes 1 $\displaystyle \frac{1}{4} x = \frac{5}{4} x$ = 120.

$\displaystyle x = 120 \times \frac{4}{5}$ = 96.

Workspace
What least number must be added to 2680 to make it a perfect square ?
• 20
• 180
• 24
• 48
• 35
Explanation

2680 lies between => 522+ > 2680 > 512

522 = 2704, therefore the number to be added is 2704 â€“ 2680 = 24.

Workspace
What is the number which when subtracted from its square is 7 times of itself ?
• 12
• 9
• 10
• 7
• 8
Explanation

Let the number be x, then x2 â€“ x = 7x. x2 = 8x => x = 8.

Workspace
The difference between the squares of two consecutive odd numbers is 48, then what are the numbers ?
• 11, 13
• 13, 15
• 15, 17
• 17, 19
• 19, 29
Explanation

Let the numbers be x and x + 2 => (x+2)2 â€“ x2 = 48.

(x2 + 4x + 4) â€“ x2 => 4x + 4 = 48 => 4x = 48 - 4 => 4x = 44, or x = $\displaystyle \frac{44}{4}$ = 11.

Therefore the numbers are 11, 13.

Workspace
The difference between the squares of two consecutive even numbers is 124, and then what are the numbers ?
• 30, 32
• 32, 34
• 28, 30
• 24, 28
• None
Explanation

Let the numbers be x and x+2

(x+2)2 â€“ x2 = 124.

Direct method = [124 â€“ 4] Ã· 4 = 30, therefore the numbers are 30 and 32.

Workspace
A worker was engaged for 30 days on the condition that he will be paid Rs.20 for every day he works, and be fined Rs.5 per day he was absent. At the end the amount received by the worker is Rs.475. For how many days the worker was absent ?
• 4 days
• 7 days
• 3 days
• 8 days
• 5 days
Explanation
If the worker worked for all the days, he would have received 30 X 20 = Rs.600
[Less] Actually received amount = Rs.475

Difference = Rs. 125 (Rs.600 - Rs.475)

On every day of absence the worker looses Rs. 20[salary] + Rs.5[fine] = Rs.25/day

Number of days of absent = $\displaystyle \frac{125}{5}$ = 5 days.

Workspace
Which of the following is a prime number?
• 787
• 767
• 707
• 717
• 737
Explanation

Answer 787, you can find up to 1000 whether a given number is prime or not by dividing it by 6.

If the remainder after dividing a number by 6 is 1 or 5 the number is a Prime.

Workspace
How many digits are used to number a book containing 580 pages?
• 1634
• 1632
• 1635
• 1636
• 1637
Explanation

Single digit 1 to 9 = 9 x 1 = 9.

Two digit 10 to 99 = 90x 2 = 180.

Three digit 100 to 580 = 481 x 3 = 1443.

Total number of digits used = 1632.

Workspace
Which number cannot be there in the units place of a perfect square ?
• 2
• 3
• 7
• 8
• All the above
Explanation
Rule: A perfect square cannot have 2,3,7,8 in its units place.
Workspace

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## Numbers Shortcuts and Tricks

0, 1, 2, 3, ..... 8, 9 are called digits.

"0" is insignificant digit and others are significant digits.

A group/set of digits is called a Numeral i.e. Number.

Place value or local value of a digit in a numeral:

In the numeral 98765432 we have.

Place value of 2 is 2 units = 2.

Place value of 3 is 3 tens = 30.

Place value of 4 is 4 hundreds = 400 and so on.

Face value: The face value of a digit in a numeral is the value of the digit it self at whatever place it may be. [i.e irrespective of its place]

Natural Numbers: The numbers which are used for counting are called natural numbers, and denoted by N. N = {1,2,3,....âˆž}.

1 is the least natural number and the greatest natural number does not exist.

Even Numbers: The integers which are exactly divisible by 2 are called even numbers. Ex: ..., -8, -6, -4, -2, 2, 4, 6, 8, 10, 12, ....âˆž i.e starting from 2 every alternate number is even.

Odd Number: The integers which are not divisible by 2 are odd numbers. Ex: 1, 3, 5, 7, 9, 11 ... âˆž.

Whole Numbers: Set of All Natural numbers including â€˜0â€™ are called whole numbers. Denoted by â€˜Wâ€™. W = {0, 1, 2, 3, ......âˆž}.

0 is the least whole number and the greatest does not exist.

Integers: Set of all whole numbers including negative numbers are called Integers, and are denoted by â€˜Iâ€™ or â€˜Zâ€™. Z = { â€¦.-3 , -2, -1, 0, 1,2,3,â€¦.âˆž}

Least and greatest integers does not exist.

Rational Number: A number which can be expressed in the form of p/q where q â‰  0 is called a rational number and denoted by Q. Ex. 1/2, 1/4, 1/3, 6/5, .....âˆž

Irrational Numbers: A number which canâ€™t be expressed in the form of p/q is called irrational number and denoted by â€™Qâ€™ Ex: $\displaystyle \sqrt{2}, \sqrt{3}, \ \sqrt{15}$, Ï€.

Real Numbers: Numbers including both rational and irrational numbers are called real numbers and denoted by â€˜Râ€™.

Prime Number: A natural number other than 1, which has factors only 1 and itself. Ex: 2,3,5,7,11,13 ......

2 is the only even prime and all the other primes are odd.

Up to 100 there are 25 prime numbers, up to 1000 there are 168 prime numbers.

Composite Numbers: Natural numbers other than prime are called composite numbers. Ex: 4,6,8,9 ......

1 is neither prime nor composite.

A composite number has at least 3 factors.

Twin primes: Twin primes which are differ by 2 are called twin primes. Ex : 3,5; / 5,7 / 11,13 / 17,19 / 29,31 / 41,43 etc.

Co- prime: Two numbers having no common factors other than 1 are called co-primes or relatively prime to each other., or HCF of that two numbers is 1. Ex: 2,3; 4,9 etc.

Perfect Number: If the sum of all the factors of a number is twice the number itself is called a perfect number. Ex: 6, 28, 496 etc.

## Test of Divisibility:

A number is divisible by:

1. By 2 => If the last digit is either 0 or even.

2. By 3 => If the sum of the digits is divisible by 3

3. By 4 => If the number so formed by the last two digits is divisible by 4.

4. By 5 => If the last digit is either 0 or 5.

5. By 6 => If it is divisible by both 2 and 3.

6. By 7 => double the units place digit and subtract it from the remaining part of the number if the difference is 0 or the difference is divisible by 7 then that number is also divisible by 7.

7. By 8 => If the number so formed by the last three digits is divisible by 8.

8. By 9 => If sum of the digits is divisible by 9.

Or if the digits of a number is reversed and subtracted from the number the difference that arrived is divisible by 9.

9. By 10 => If the last digit is Zero.

10. By 11 => If the difference of sums of the alternate digits of the number is either â€˜0â€™ or a multiple of 11.

Or => 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 â€˜0â€™ or a number divisible by 11.

11. By 12 => Any number which is divisible by both 3 and 4, is also divisible by 12.

12. By 14 => Any number which is divisible by both 2 and 7, is also divisible by 14.

13. By 15 => Any number which is divisible by both 3 and 5, is also divisible by 15.

## Numbers Important Formulas

Sum or difference of two even numbers is Even.

[Even â€“ Even = Even] or [Even + Even = Even].

Ex: 8 - 18 = 10, 8 + 18 = 26 [both are Even].

Sum or difference of two Odd numbers is Even.

[Odd + Odd = Even] [Odd â€“ Odd = Even].

[15 â€“ 5 = 10] [33 â€“ 15 = 18] both 10, 18 are even.

Sum or difference of one Even one Odd is Odd.

[odd + even = odd] ex: 4 + 5 = 9 [odd].

Product of two even numbers is Even.

Even x Even = Even. Ex: 8 x 8 = 64.

Product of two odd numbers is Odd.

Odd x Odd = Odd. Ex: 9 x 9 = 81.

Product of one even and one odd is Even.

Even x Odd = Even, Ex: 8 x 15 = 120.

Sum of difference between one Odd and one Even is Odd.

[Odd + Even = Odd] [Odd â€“ Even = Odd].

## Rule of Divisibility:

Let â€˜aâ€™ and â€˜bâ€™ are two integers and if â€˜aâ€™ is exactly divisible by â€˜bâ€™ then b is a factor of â€˜aâ€™, and, if a is multiplied with c then â€˜acâ€™ is also divisible by â€˜bâ€™

The number to be divided is called Dividend, the number that divides the other number is called Divisor, and the result obtained is called Quotient and un divided part is called Remainder.

Dividend = Divisor x Quotient + Remainder.

Divisor = $\displaystyle \frac{\text{Dividend} - \text{Remainder}}{\text{Quotient}}$

## Arithmetic Progressions:

Progression is sequence of numbers, and each term differs by a constant term [d] â€˜nâ€™th term = Tn = a + (n â€“ 1).

Sum of the terms Sn = $\displaystyle \frac{n}{2} \left(2a + (n â€“ 1)\right)$ or $\displaystyle \frac{n}{2} \left(\text{First term} + \text{First term}\right)$

Sum of Natural numbers = $\displaystyle \frac{n(n + 1)}{2}$

Ex: sum of 1,2,3,4,5,6 = n = 6 sum = $\displaystyle \frac{6(6 + 1)}{2} = \frac{42}{2}$ = 21.

Sum of Even numbers = n( n + 1).

Ex: sum of 2,4,6,8,10 = n = 5, sum = 5(5 + 1) = 5 x 6 = 30.

Sum of Odd numbers = n2

Ex: sum of 1,3,5,7,9,11 = n= 6, then sum = 62 = 36.

Sum of the squares of Natural numbers = (12 + 22 + 32 + .... + n2) = $\displaystyle \frac{n(n + 1)(2n + 1)}{6}$

Sum of the Cubes of Natural numbers = (13 + 23 + 33 + ... + n3) = $\displaystyle \frac{n^2(n + 1)^2}{4}$

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