## Numbers Questions - 1

How many numbers are divisible by both 2 and 3 up to 300?
• 50
• 100
• 150
• 200
• 250
Explanation

Divide 300 by 6, as the numbers which are divisible by both 2 and 3 are also divisible by 6.

The quotient obtained after dividing 300 by 6 is 50, hence answer is 50.

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How many numbers are divisible by 3 and 4 up to 200?
• 116
• 16
• 100
• 84
• None
Explanation

Divide 200 by 12, as the numbers which are divisible by both 3 and 4 are also divisible by 12.

The quotient obtained after dividing 200 by 12 is 16.

So 16 numbers are the common multiples of 3 and 4.

Now the number of multiples of 3 up to 200 is 66, and of 4 is 50.

The numbers divisible by both 3 and 4 is [66 + 50 – 16] = 100.

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The sum of five consecutive odd numbers is greater than its middle number by 76. What is the middle number ?
• 15
• 17
• 19
• 21
• 23
Explanation

Let the first number be x, then the other numbers are x, x + 2, x + 4, x + 6, x + 8.

Now sum of the numbers is (5x + 20 ), middle number is (x + 4).

Then (5x + 20 ) – (x + 4) = 76 => 4x + 16 = 76 => 4x = 76 – 16 => 4x = 60 => x = $\displaystyle \frac{60}{4}$ = 15.

The numbers are 15, 17, 19, 21, 23; middle number is 19.

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$\displaystyle \frac{1}{3}$rd of a number exceeds its $\displaystyle \frac{1}{4}$th by 25, then $\displaystyle \frac{1}{2}$ of the number is ?
• 300
• 150
• 125
• 175
• 100
Explanation

Let the number be x then $\displaystyle \frac{1}{3} x - \frac{1}{4} x$ = 25 => $\displaystyle \frac{1}{12} x$ = 25 or x = $\displaystyle \frac{12}{1} \times$ 25 = 300.

300 $\displaystyle \times \frac{1}{2}$ = 150.

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A boy was asked to multiply a number by $\displaystyle \frac{2}{3}$ but he multiplied the number by $\displaystyle \frac{3}{2}$ and thereby he got 30 more than the correct answer. What is the number ?
• 48
• 45
• 36
• 63
• 75
Explanation

Let the number be x, then $\displaystyle \frac{2}{3} x – \frac{3}{2} x = 30 => \frac{4x - 9x}{6}$ = 30 => 5x = 6 x 30.

$\displaystyle x = \frac{180}{5}$ = 36, hence the number is 36.

Workspace
By which number 424 can be divided to get a perfect square ?
• 24
• 100
• 103
• 106
• 12
Explanation

Standard form/Product of prime factors of 424 = 23 x 53 = 22 x 21 x 531

Therefore 424 be divided by 106 [2 x 53] => $\displaystyle \frac{424}{106}$ = 4 = 22

Workspace
A number when divided by 36 leaves a remainder 10. What will be the remainder if the same number is divided by 9 ?
• 1
• 2
• 3
• 4
• 5
Explanation

Let the quotient be Q, then the number is 36Q + 10.

Here we have two divisors D1 = 36, D2 = 9, D2 must be a factor of D1.

As 9 is a factor of 36, then 36Q is also divisible by 9, Now divide 10 by 9, thus the remainder will be 1.

Workspace
A number when divided by 24 leaves a remainder 4. What will be the remainder if the square of the number is divided by 24 ?
• 20
• 30
• 16
• 12
• 18
Explanation

As the both the divisors are same then directly take the square of the remainder and divide it by 24, 42 = 16, as 16 is not divisible by 24 then the remainder will be 16 only.

Workspace
What is the units in the product 288287 ?
• 2
• 8
• 6
• 4
• None
Explanation

Units digit of 288 is 8 then the units digit of the product will the one of the power cycle of 8

8 has 4 power cycles => 8,4,2,6 respectively 84Q + 3, therefore the 3rd power of 8 will be the units Digit of the product of 288287 = units digit = 2

Workspace
What is the units digit in the product 2326 x 2929 ?
• 1
• 2
• 3
• 4
• 5
Explanation

Units digit of 23 is 3, and 3 has 4 power cycles = 3,9,7,1

Then 2326 = 34Q + 2, 2nd power cycle of 3 is 9,

Units digit of 29 is 9 , 9 has 2 power cycles => 9, 1

Then 2929 = 92Q +1 = first power cycle of 9 is 9

So 9 x 9 = 81, thus the units place of the product will be 1.

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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,....∞}.

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, ....∞ 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 ... ∞.

Whole Numbers: Set of All Natural numbers including ‘0’ are called whole numbers. Denoted by ‘W’. W = {0, 1, 2, 3, ......∞}.

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 ‘I’ or ‘Z’. Z = { ….-3 , -2, -1, 0, 1,2,3,….∞}

Least and greatest integers does not exist.

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

Irrational Numbers: A number which can’t be expressed in the form of p/q is called irrational number and denoted by ’Q’ Ex: $\displaystyle \sqrt{2}, \sqrt{3}, \ \sqrt{15}$, π.

Real Numbers: Numbers including both rational and irrational numbers are called real numbers and denoted by ‘R’.

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 ‘0’ 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 ‘0’ 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 – 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 – Odd = Even].

[15 – 5 = 10] [33 – 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 – Even = Odd].

## Rule of Divisibility:

Let ‘a’ and ‘b’ are two integers and if ‘a’ is exactly divisible by ‘b’ then b is a factor of ‘a’, and, if a is multiplied with c then ‘ac’ is also divisible by ‘b’

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] ‘n’th term = Tn = a + (n – 1).

Sum of the terms Sn = $\displaystyle \frac{n}{2} \left(2a + (n – 1)\right)$ 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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