Study Notes for CAT 2021: Remainder (Number System)

Basic Formula of Number System:

1. Sum of all the first n natural numbers =

For example:  1+ 2 +3 +…..+105= 

2. Sum of first n odd numbers =

For example 1+3+5+7==16(as there are four odd numbers)

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

For example : 2+4+6+8+….+100 (or 50^th even number) = 50×(50+1)= 2550

4. Sum of squares of first n natural numbers =

For example:

5. Sum of cubes of first n naturals numbers =

For example:

Example:

(1) What is the total of all the even numbers from 1 to 400?

Solution:

From 1 to 400, there are 400 numbers. So, there are 400/2= 200 even numbers.

Hence, sum = 200(200+1) = 40200     (From Rule III)

(2) What is the total of all the even numbers from 1 to 361?

Solution:

From 1 to 361, there are 361, there are 361 numbers; so there are even numbers. Thus, sum = 180(180+1)=32580

(3) What is the total of all the odd numbers from 1 to 180?

Solution:

Therefore are 180/2 = 90 odd numbers between the given range. So, the sum =

(4) What is the total of all the odd numbers from 1 to 51?

Solution

There are odd numbers between the given range. So, the sum =

(5) Find the of all the odd numbers from 20 to 101.

Solution:

The required sum = Sum of all the odd numbers from 1 to 101.

Sum of all the odd numbers from 1 to 20

= Sum of first 51 odd numbers – Sum of first 10 odd numbers

=

Miscellaneous

1. In a division sum, we have four quantities – Dividend, Divisor, Quotient and Remainder. These are connected by the relation.

Dividend = (Divisor × Quotient) + Remainder

2. When the division is exact, the remainder is zero (0). In this case, the above relation becomes

Dividend = Divisor × Quotient

Example: 1: The quotient arising from the divisor of 24446 by certain numbers is 79 and the remainder is 35; what is the divisor?

Solution:

Divisor × Quotient = Dividend -  Remainder

79×Divisor = 24446 -35 =24411

Divisor = 24411 ÷ 79 = 309.

Example: 2: A number when divided by 12 leaves a remainder 7. What remainder will be obtained by dividing the same number by 7?

Solution:

We see that in the above example, the first divisor 12 is not a multiple of the second divisor 7. Now, we take the two numbers 139 and 151, which when divided by 12, leave 7 as the remainder. But when we divide the above two numbers by 7, we get the respective remainder as 6 and 4. Thus, we conclude that the question is wrong.

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