Numbers Formulas

Numbers Formulas

1. Prime Number:

A Prime Number is a number which has factors only unity and itself.

Example:2,3,5,7..........

2. Composite Number:

A number which has more than two different factors is called a Composite Number.

Example:4,6,8,10...........

3. Co-Prime Number:

Two numbers are said to be Co-Prime to each other of they do not have any common factor other than one.

Example:(2,3)(5,7)...........

4. Twin Prime Number:

Two prime numbers are said to be twin prime if the difference between them is two.

Example:(3,5)(5,7),(11,13)...........

5. Perfect Number:

A number is said to be perfect number if the sum of all the positive integral divisors of the numbers including unity and itself is equal to twice the number.

If 'a' is even,2^a-1(2 power of a) is always divisible by 3
If 'a' is odd, 2^a+1(2 power of a) is always divisible by 3
If 'a' is odd, 2^2a+1(2 power of 2a) is divisible by 5
If 'a' is even, 2^2a-1(2 power of 2a) is divisible by 5
If 'a' is odd,5^2a+1(5 power of 2a) is divisible by 13
If 'a' is even 5^2a-1(5 power of 2a) is divisible by 13
If a<b then a<(a+b/2)<b,because(a+b/2) is the average of 'a' and 'b'.

7.The standard form of writing a number is a*10^n(10 power of n) Where 'a' lies between 1 and 10 and 'n' is an integer-Positive and negative.

Example:325600=3.256*10^5 and 0.3256=3.256*10^-1(10 power of -1).

8. i.x^a+y^a(x power of a and y power of a) is divisible b x+y When 'a' is odd
x^a+y^a=(x+y)(x^a-1-yx^a-2 +......+y^a-1)(x power of a-1)
ii.x^a-y^a is divisible by X+y When 'a' is even
x^a-y^a=(x+y)(x^a-1-yx^a-2+.........+xy^a-2-y^a-1)
iii.x^a-y^a is always divisible by x-y
x^a-y^a=(x-y)(x^a-1+yx^a-2+........+y^a-1)

9.Tests for Divisibility:

i. A number is divisible by 2,whene its unit digit is even or zero

Example:262,240,246....

ii. A number is divisible by 3,when the sum of its digits is divisible by 3.

Example:156,27

iii. A number is divisible by 4, when the number formed by the last 2 right hand digits is divisible by 4 or if the last two digits are zero.

Example:144,1200.

iv. A number is divisible by 5, when its units digit is 5 or 0

example:1245,1350

v. A number is divisible by 6,when its divisible by 2 and 3 both.

example:126,78

Vi. No rule is still known for divisibility of a number by 7.

Vii. A number is divisible by 8,when the number formed by the right hand digits is divisible by 8 (or)
last three digits are zeros.

Example:4408,1200

Viii. A number is divisible by 9,When the sum of the digits is divisible by 9.

example:1926,1359

ix. A number is divisible by 10, when its unit digit is 0.

Example:4000,1850

x. A number is divisible by 11, when the differece between the sum of the digits in the odd and even places is zero or a multiple of 11.

Example:91531,2332

xi. A number is divisible by 12, when it is divisible by 3 and 4 both.

Example:132,2556

xii. A number is divisible by 25, when the number formed by the last two right hand digits is divisible by 25.

Example:143575,12150.

xiii. A number is divisible by 125, when the number formed by the last three right hand digits is divisible by 125.

Example:139625,460250.

xiv. For any integer 'a'.
a^3-a is divisible by 3
a^5-a is divisible by 5
a^7-a is divisible by 7
a^11-a is divisible by 11
a^13-a is divisible by 13.
In general, if p is prime numbe then for any whole number 'a' a^p-a is divisible by P.

xv.When any number with even number of digits is added to its reverse, the sum is always multiple by 11.

Example:5762+2675=8437 which is divisible by 11 .

Note:A number is divisible by another number if the former is divisible by all the factors of the later.

10. Sum of first 'n' natural numbers=n(n+1/2)

11. Sum of first 'n' odd numbers=n^2

12. Sum of first 'n' even natural numbers=n(n+1)

13. In division sum we have dividend,divisor,quotient and remainder.
Dividend=Divisor*Quotient+Remainder
(or)
Divisor=Dividend-Remainder/Quotient

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