Number System - Rules and Examples

Rules on Counting Numbers

Note:

In the first n counting numbers, there are n/2 odd and n/2 even numbers provided n, the number of numbers, is even. If n, the number of numbers, is odd, then there are 1/2(n + 1) odd numbers and 1/ 2 (n – 1) even numbers.

For example:  from 1 to 50, there are 50/2= 25 odd numbers and 50/2 = 25 even numbers. And from 1 to 51, there are (51+1 )/2 = 26 odd numbers and(51-1 )/2= 25 even numbers.

 The difference between the squares of two consecutive numbers is always an odd numbers.

 The difference between the square of two consecutive numbers is the sum of the two consecutive numbers.

Solved examples:

Ques 1. What is the total of all the even numbers from 1 to 400?

Ques 2. What is the total of all the odd numbers from 1 to 180?

Ques 3. Find the sum of all the odd numbers from 20 to 101.

POWER AND INDEX

If a number ‘p’ is multiplied by itself n times, the product is called nth power of ‘p’ and is written as pⁿ. In pⁿ, p is called the base and n is called the index of the power.

Solved examples:

Ques 4. What is the number in the unit place in (729)⁵⁹?

Solution: When 729 is multiplied twice, the number in the unit place is 1. In other words, if 729 is multiplied an even number of times, the number in the unit place will be 1. Thus, the number in the unit place in (729)⁵⁸ is 1.

            So, (729)⁵⁹ = (728)⁵⁸ X (729) = (…….1) X(729) = 9 in the unit place.

Note: When you solve this type of questions (for odd numbers) try to get the last digit 1, as has been done in the above example.

Ques 5. Find the number in the unit place in (98)⁴⁰ , (98)⁴² , (98)⁴³.

Solution:       (98)⁴ = (……….6)

            So,       (98)⁴ⁿ =(…………6)

Thus,   (98)⁴⁰ = (98)^4x10 = (…..6) = 6 in the unit place.

            (98)⁴² = (98)^4x10 X (98)² = (…..6)X(…….4) = 4 in the unit place.

            (98)⁴³ = (98)^4x10 X(98)³ = (…..6)X(…….2) = 2 in the unit place.

Note: When there is an even number in the unit place of base, try to get 6 in the unit place, as has been done in the example.

Rule no. 1.

For Odd Numbers

When there is an odd digit in the unit place (except 5), multiply the number by itself until you get 1 in the unit place.

          

  (…………1) ⁿ = (……………1)

   (……….3)⁴ⁿ   = (………..1)

   (……...7)⁴ⁿ     =  (………..1)

            

Where n = 1, 2, 3,……….  

Rule no.2

When there is an even digit in the unit place of the given number, then after any times of its multiplication, it will have the same digit in the unit place i.e.

            (…………1) ⁿ = (……………1)

            

               (……….5)ⁿ = (………..5)

                 (……...6)ⁿ =(………..6)

Ques 6. What is the numbers in the unit place when 781, 325, 497 and 243 are multiplied together?
Solution: Multiply all the numbers in the unit place

 i.e., 1 X 5 X 7 X 3; the result is a number in which 5 is in the unit place.