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 nonnegative 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 numbers 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.

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*5^{0  +} 2⁰*3⁰*5^{0 +} 2⁰*3¹*5^{1 +} 2⁰*3¹*5^{1 +} 2¹*3⁰*5⁰ + 2¹*3⁰*5⁰ + 2¹*3¹*5¹ + 2¹*3^1*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

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 3^rd equation remaining terms are all 0’s.

Finding the highest power of a prime 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 4^th 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.