Maths Tricks for Problems based on Numbers

Important Rules on Counting Numbers:-

Rule 1: Sum of first n natural numbers :
                              Σ_n  = {n × (n + 1)}/ 2

Rule 2: Sum of first n odd numbers :

                               = n × n = n²

Rule 3: Sum of first n even numbers = n(n+1).
Rule 4: Sum of squares of first n natural numbers : 

                              Σ_n²  = {n × (n  + 1) × (2n + 1)}/ 6

Rule 5: Sum of cubes of first n natural numbers :

                               Σ_n³ = [n × (n + 1)/ 2]²
Rule 6: If n is the number of numbers and n is even then n/2 numbers will be even and n/2 numbers will be odd among first n natural numbers.

Rule 7: If n is odd , then there are (n+1)/2 odd numbers and (n-1)/2 even numbers.

Rule 8: The difference between the squares of two consecutive numbers is always an odd number.

29² – 28² = 57

Rule 9: The difference between the squares of two consecutive numbers is the sum of the two consecutive numbers.

Ex. In the above example 29 + 28 = 57.

Ex. Find out the number of all even numbers from 1 to 500?

Solution : Since 300 is an even number so total number of even numbers will be (n/2)

Number of Even numbers= (500/2) = 250 even numbers.

Ex. What is the sum of all the even numbers from 1 to 481

Solution : Even numbers will be = (481-1)/2= 240. 
Sum of even numbers = n × (n+1) = 240 (240+1) = 57,840.

Ex. Find out of sum of all the odd numbers from 80 to 250 

Solution : Required Sum =Sum of all odd numbers from 1 to 250 - Sum of all odd numbers from 1 to 80 :

Required Sum of the odd numbers = 125² – 40² = 14,025.

Rule 10 : Dividend = (Divisor × Quotient) + Remainder

Ex. What least number must be added to 7993 to make it exactly divisible by 75 ? 

Solution : On dividing 7993 by 75 we get 43 as remainder,

So the number to be added will be 75 - 43 = 32 to make it perfectly divisible by 75.

Ex. What least number must be subtracted from 7983 to make it exactly divisible by 75 ?

Solution : On dividing 7983 by 75 we get 33 as remainder,

So the number to be subtracted will be 33.

Ex. Find the least number of five digits which is exactly divisible by 83 ?

Least number of five digits will be 10000, on dividing 10000 by 73 we get 40 as remainder,

so the number to be added will be = 83 - 40 = 43

So the required least 5-digit number will be = 10000 + 43 = 10043.

Rule : For finding least number first subtract the remainder from the divisor and then add the calculated difference to the least number (10000) to get the required least five digit number.

Ex. find the greatest number of five digits which is exactly divisible by 97 ?

Solution : The greatest number of five digit will be 99999, on dividing it by 97 we get 89 as remainder,

so the required number will be 99999 - 89 = 99910 

Rule : For finding greatest number substract the remainder to the greatest number (99999).

Hope you have enjoyed the base concept of this chapter till now. More Conceptual enjoyment with the Various Questions related to this chapter which are frequently asked in the examinations will be presented to your for self study on this blog soon, just keep visiting this blog regularly and writing me your related queries through comment or forum. Best of Luck for Exams. To be continued soon........................