FORMULAE


Number System


Real numbers are of two types: Decimal and Integers. The Integers have no decimal values like 0.33, 0.45; Decimals can be 0.23, 0.333... etc


Integers are of three types negative numbers like -1, -2 .. , zero and positive numbers like 1, 2, 3...


The negative numbers and zero are called non positive and zero and positive numbers are called non negative numbers.




Decimal numbers are of finite or terminating decimal types or infinite decimal types. The infinite decimal type is classified as rational numbers if they can be expressed in the form p / q or irrational number if they can't be expressed in the form p / q.




Properties:


  1. The remainder on division of a prime number p > = 5 by 6 is 1 / 5.
  2. The remainder of the division of a square of a prime number p > = 5 by 12 / 24 is 1.
  3. If a and b are odd prime numbers a2 - b2 and a2 + b2 is composite.


HCF of 'x' and 'y' is G then HCF of x , (x+y) and x , (x-y) and (x+y) , (x-y) is also G.




Number Theory
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Theorem of Divisibility


Calculate the Sum of factors of a number:


Step 1:  Get prime factors of a number say 240


240 = 24 * 31 * 51


Step 2: Sum of factors formula is


240 = (20 + 21 + 22 + 23 + 24) * (30 + 31) * (50 + 51)


Step 3: 31*4*6 = 744



Calculate the Number of factors of a number:


Step 1: Get the prime factors of a number


240 = 24 * 31 * 51


Step 2: Number of factors of a number.


Number of factors = ( 4 + 1 ) * ( 1 + 1) * ( 1 + 1) = 5 * 2 * 2 = 20

Thus the powers of the numbers are increased by one and multiplied.



Calculate the sum and number of even factors of a number:


Step 1: Get the prime factors of a number




240 = 24 * 31 * 51


Step 2: Sum of even factors
Sum = (21 + 22 + 23 + 24) * (30 + 31) * (50 + 51)
Number of even factors = 4 * 2 * 2 = 16
Thus the powers of the numbers are increased by one and multiplied except 2.

Calculate the sum and number of odd factors of a number:

Step 1: Get the prime factors of a number




240 = 24 * 31 * 51


Step 2: Sum of odd factors
Sum = (20) * (30 + 31) * (50 + 51)
Number of odd factors = 1 * 2 * 2 = 4
Thus the powers of the numbers are increased by one and multiplied except 2.

Calculate the sum and number of factors of a number satisfying a condition:

Step 1:  Get prime factors of a number say 240




30 = 21 * 31 * 51


Step 2: Sum of factors formula is


30 = (20 + 21 ) * (30 + 31) * (50 + 51)


which is expanded as
= 20*30*50  + 20*30*50 + 20*31*51 + 20*31*51 + 21*30*50 + 21*30*50 + 21*31*51 + 21*31*51

These are all the factors of the number, We can apply any condition we want and remove unnecessary of them viz. remove factors that are perfect squares, not perfect squares, between higher and lower limit etc




Divisibility theory
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Highest Power in a factorial


Finding the number of zeroes in a factorial:


Step 1: Select a number and use the formula


Find zeroes in 127! = [127 / 5] + [127 / 52] + [127 / 53] + [127 / 54] ...

                                 = 25 + 5 + 1 + 0 = 31




Remember that we ignore decimal values so after 3rd equation remaining terms are all 0's.


Finding the highest power of a number in a factorial:


Step 1: Select a number and use the formula


Find highest power of 3 in 127! = [127 / 3] + [127 / 32] + [127 / 33] + [127 / 34] + [127 / 35] + ...

                                 = 42 + 14 + 4 + 0 = 60


Remember that we ignore decimal values so after 4th equation remaining terms are all 0's.




Finding the highest power of a composite number in a factorial:


Step 1: Select a number and use the formula


Find highest power of 15 in 127! but 15 is composite so prime factors are 3 * 5.


Step 2: Find highest power of each prime factor.


Find highest power of 3 in 127! = [127 / 3] + [127 / 32] + [127 / 33] + [127 / 34] + [127 / 35] + ...

                                 = 42 + 14 + 4 + 0 = 60


Find highest power of 5 in 127! = [127 / 5] + [127 / 52] + [127 / 53] + [127 / 54] ...

                                 = 25 + 5 + 1 + 0 = 31


Choose the lesser of both values and that is the answer = 31.




Factorial and Highest power
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