Number System Part-2

Number System Part-2   

Important Rules on Counting:-

Rule 1: Sum of first n natural numbers :

                              

Rule 2: Sum of first n odd numbers :

                              

Rule 3: Sum of first n even numbers = n(n+1)

Rule 4: Sum of squares of first n natural numbers :

                              

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

                              

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

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



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

Ex. In the above example 9 + 8 = 17

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Ex. Find out the number of all even numbers from 1 to 300 ?

Solution : Since 300 is an even number so total number of even numbers will be (n/20) = (300/2) = 150 even numbers

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Ex. What is the sum of all the even numbers from 1 to 381

Solution : Even numbers will be = (381-1)/2= 190
Sum of even numbers = n * (n+1) = 190 (190+1) = 36,290

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Ex. Find out of sum of all the odd numbers from 50 to 200

Solution : Required Sum = Sum of all odd numbers from 1 to 200 - Sum of all odd numbers from 1 to 50 :

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

Ex. What least number must be added to 7963 to make it exactly divisible by 65 ?

Solution :On dividing 7963 by 65 we get 33 as remainder, So the number to be added will be 65 - 33 = 32

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Ex. What least number must be subtracted from 7963 to make it exactly divisible by 65 ?

Solution : On dividing 7963 by 65 we get 33 as remainder, So the number to be subtracted will be 33

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Ex. Find the least number of five digits which is exactly divisible by 73 ?

Least number of five digits will be 10000, on dividing 10000 by 73 we get 72 as remainder, so the number will be = 10000 + 72 = 10072

Rule : for finding least number add the remainder to the least number

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Ex. find the greatest number of five digits which is exactly divisible by 147 ?

Solution : The greatest number of five digit will be 99999, on dividing it by 147 we get 39 as remainder, so the required number will be 99999 - 39 = 99960

Rule : For finding greatest number subtract the remainder to the greatest number