# Quantitative Aptitude - Formula

#### Prime Number and its properties

A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

• The smallest prime number is 2 and it is the only even prime number.
• The smallest odd prime number is 3.
• Prime Number between 1 and 100 are
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97
• Prime Number between 100 and 200 are
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199
• If a and b are two odd prime numbers then a2 - b2 & a2 + b2 both will be composite.
• The remainder of division of square of a prime number p>=5 divided by 12 is 1.
• The remainder of division of square of a prime number p>=5 divided by 24 is 1.

#### Composite Numbers

Numbers greater than 1 which are not prime, are known as composite numbers. For example: 4, 6, 8, 9 etc. These numbers are divisible by numbers other than 1 and number itself.

#### Basic Number System Formula

$${(a+b)^2=a^2+b^2+2ab}$$

$${(a−b)^2=a^2+b^2−2ab}$$

$${(a+b)^2−(a−b)^2=4ab}$$

$${(a+b)^2+(a−b)^2=2(a^2+b^2)}$$

$${(a^2−b^2)=(a+b)(a−b)}$$

$${(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)}$$

$${(a^3+b^3)=(a+b)(a^2−ab+b^2)}$$

$${(a^3−b^3)=(a−b)(a2+ab+b2)}$$

$${(a^3+b^3+c^3−3abc)=(a+b+c)(a^2+b^2+c^2−ab−bc−ca)}$$

$${If a+b+c=0, then a^3+b^3+c^3=3abc}$$

#### Square of a number n

n2= (n-1)2 + (n-1) + n

#### Rules for Finding out HCF and LCM of fraction

HCF of given fractions = ${\frac{HCF \, of \, Numenators}{LCM \, of \, Numenators}}$

LCM of given fractions = ${\frac{LCM \, of \, Numenators}{HCF \, of \, Numenators}}$

#### Relation between HCF and LCM

If a & b are two numbers then,

H.C.F of (a,b) x L.C.M of (a,b) = a x b

#### H.C.F of two numbers having exponents

The H.C.F of (am - 1)(an - 1) is given by (aH.C.F of m, n - 1)

#### Divisibility Rules

Divisibility By 2

A number is divisible by 2, if its unit's digit is any of 0,2,4,6,8

Divisibility By 3

A number is divisible by 3, if the sum of its digits is divisible by 3

Divisibility By 4

A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Divisibility By 5

A number is divisible by 5, if its unit's digit is either 0 or 5.

Divisibility By 6

A number is divisible by 6, if it is divisible by both 2 and 3.

Divisibility By 8

A number is divisible by 8, if the number formed by the last Three digits of the given number is divisible by 8.

Divisibility By 9

A number is divisible by 9, if the sum of its digits is divisible by 9.

Divisibility By 10

A number is divisible by 10, if it ends with 0.

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 number divisible by 11.

Divisibility By 12

A number is divisible by 12, if it is divisible by both 4 and3.

#### Last two digit of any multiplication operation

To find last two digit of any multiplication i.e a x b x c x d.... multiply last two digit of a x b with c and so on...

#### Important Outcomes using Divisibility Rules

• Any number which is written in the form of 10n - 1 is divisible by 3 and 9.
• If m and n are integers then (m+n)! is divisible by m!n!
• Product of Number and its outcome

Odd x Odd = Odd
Even x Even = Even
Even x Odd = Even
Odd x Even = Even

• The difference between two numbers (xy) - (yx) is divisible by 9
• n3 - n is divisible by 6
• ${\frac{(a+1)^n}{a}}$ will always give remainder of 1.
• ${\frac{(a)^n}{({a+1})}}$ leaves remainder 1 if 'n' is even and 'a' as remainder if 'n' is odd.

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