Number System - 3

Division and Remainder

When a given number is not exactly divisible by any number, then there is a remainder number at the end of such division.

Suppose we divide 25 by 7 as,

Then, divisor =7, dividend = 25, quotient = 3 and remainder = 4.

There is most important formula to check the divisibility:-

Dividend = (divisor x quotient) + remainder

So, if a number ‘x’ is divided by ‘k’, leaving remainder ‘r’ and giving quotient ‘q’ then the number can be found by using (i)

x = kq + r

Hence, if the number x is exactly divisible by k, then remainder = r = 0

x = kq

and so ^x/_k = q, implying that x is divisible exactly by ‘k’ and ‘q’ is an integer.

Methods to Find a Number Completely Divisible by Another

Consider a given number x. when divided by ‘d’, it gives a quotient ‘q’ and remainder ‘r’.

Now, to find a number exactly divisible by ‘d’, we can use either of the following two methods to reduce the remainder to zero. ( If a number is exactly divisible, then remainder is zero).

Method 1

By subtracting the remainder from the given number ( dividend).

The required number that is exactly divisible by ‘d’ = x – r

Hence ‘remainder’ is the least number that can be subtracted from any number to make it exactly divisible.

Method 2

By adding the (divisor – remainder) to the given number.

The required number that is exactly divisible by d = x + (divisor – remainder)

Therefore, ( divisor – remainder ) is the least number that can be added to any given number to make it exactly divisible.

Example:

Find the least number, that must be

(a) Subtract from, or

(b) Added to a given number 5029, to make it exactly divisible by 17.

Soln.:- On dividing 5029 by 17, we get remainder = 14.

(a) The least number to be subtracted to make it exactly divisible = remainder = 14.

(b) The least number to be added to make it exactly divisible = divisor – remainder = 17 – 14 = 3.

Greatest n-digit and Least n-digit number Exactly Divisible by a number

(a) To find out the greatest n –digit number exactly divisible by a divisor ‘d’, we use method 1.

The required number = greatest n-digit number – remainder.

(b) To find out the least n-digit number exactly divisible by a divisor ‘d’, we use method 2, because if we use method 1, then subtracting any number from the n-digit least number will reduce it to (n-1) digit number.

The required number = least n-digit number + (divisor – remainder)

Example:- Find the

(a) greatest 3-digit number divisible by 13.

(b) The least 3-digit number divisible by 13.

Soln.:-

(a) Greatest 3-digit “999” divided by 13, we get remainder = 11

So, the required 3-digit greates number = 999 – 11 = 988

(b) Least 3-digit “100”divided by 13, we get remainder = 9

So, the required 3-digit least number = 100 + (13 – 9) = 104.

Number System - 2

After types of number system, we study about divisibility property of numbers.

There are certain tests for divisibility of numbers by any of the numbers 2, 3, 4, 5, 6, 8, 9, 10 and 11 such that by simply examining the digits in the given number, one can easily determine whether or not a given number is divisible by any of these numbers.

Some test detailed as follows:

Divisibility by 2
If the last digit is an even number or it has zero (0) at the end.
Ex.:- 74, 148, 1210 are all divisible by 2.

Divisibility by 3
If the sum of the digits of the given number is divisible by 3, then that number is fully divisible by 3.
Ex.:- The sum of the digits of number 3705 is 3 + 7 + 0 + 5=15. 15 is divisible by 3 so the number 3705 is fully divisible by 3.

Divisibility by 4
If the formed by the last two digits of the given number is divisible by 4, or if the last two digits are ‘00’, then that number is fully divisible by 4.
Ex.:-216560 is a number whose last two digits are 60, 60 is divisible by 4 so the given number 216560 is also divisible by 4.

Divisibility by 5
If the last digit of the given number is 0 or 5 then the number is fully divisible by 5.
Ex.:- 865, 1605, 5670 are all divisible by 5.

Divisibility by 6
If the given number is divisible by 2 and 3 then that number is also divisible by 6.
Ex.:- let us consider the number is 89004. It has 4 as the last digit, so it is divisible by 2.
Sum of the digits 8 + 9 + 0 + 0 + 4=21, 21 is divisible by 3.
Since, 89004 is divisible by 2 and 3 both so that number is also divisible by 6.

Divisibility by 8
If the number formed by the last three digits of the given number is divisible by 8 or if the last three digits are ‘000’, then that number is fully divisible by 8.
Ex.:- The number 56976 has 976 as the last three digits. Since 976 is divisible by 8, 56976 is also divisible by 8. The number 5463000 end with ‘000’ and so, it is divisible by 8.

Divisibility by 9
If the sum of the digits of the given number is divisible by 9, then that number is also fully divisible by 9.
Ex.:- 890676 is a number the sum of whose digits is = 8 + 9 + 0 + 6 + 7 + 6=36. Since 36 is divisible by 9 so the number 890676 is also divisible by 9.

Divisibility by 10
If the last digit of the number is zero (0), then that number is fully divisible by 10.
Ex.:- 890 has 0 at the end, so it is divisible by 10.

Divisibility by 11
If the difference of the sum of its digits in odd places and the sum of its digits in even places is either zero (0) or a multiple of 11.
Ex.:- Let us consider the number 647053.
Sum of digits at odd places = 6 + 7 + 5= 18
Sum of digits at even places = 4 + 0 + 3= 7
Difference of the sums = 18 – 7=11
Since the difference 11 is a multiple of 11, so 647053 is divisible by 11.

Some General properties of Divisibility :-

1. If a number ‘x’ is divisible by another number ‘y’, then any number divisible by ‘x’, will also be divisible by ‘y’ and by all the factors of ‘y’.
Ex.:- The number 84 is divisible by 6. Thus any number that is divisible by 84, will also be divisible by 6 and also by the factors of 6, i.e. by 2 and by 3.

2. If a number ‘x’ is divisible by two or more than two co-prime number then x is also divisible by the product of those numbers.
Ex.:- The number 2520 is divisible by 5, 4, 13 that are prime to each other (i.e. co-prime), so, 2520 will also be divisible by 20 (= 5 x 4), 52 (= 4 x 13).

3. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their sum x + y is also divisible by ‘p’.
Ex.:- The number 225 and 375 are both divisible by 5. Thus their sum = 225 + 375 = 600 will also be divisible by 5.
Note:- It is also true for more than two numbers.

4. If two number ‘x’ and ‘y’ are divisible by a number ‘p’, then their difference x – y is also divisible by ‘p’.
Ex.:- The number 126 and 507 are both divisible by 3. Thus their difference = 507 – 126 = 381 will also be divisible by 3.

Number System - 1

In aptitude we first start with the Number system.

Here, numbers are divided in some category i.e. Natural number, Whole number, Integer, rational number, irrational numbers etc.

Natural numbers:- It is start from 1 to infinity and it is non-negative numbers.
Set of natural numbers is {1, 2, 3, ……}

Whole number:- It is also non-negative number including “0”.
Natural number including “0” is called Whole Numbers.
Set of whole numbers is {0, 1, 2, 3, …….}

Integers:- All numbers are called integer it may be positive or negative.
Set of Integers is {……,-3, -2, -1, 0, 1, 2, 3,……}

Rational number:- Any number that can be expressed as the quotient a/b of two integers, with the denominator b not equal to zero is called Rational number.
Set of rational number is ½, 3/5, -8/5, 0, +3, -150, …….

Irrational number:- Any real numbers are called Irrational number which is not a Rational number.
Set of irrational number is π, √2,√5,√7, …….

Some numbers are like √-8,√-7,√-5,√-1,3+√-7 and so on. These are undefined numbers, called Complex numbers.

Integers- There are three types of integers :-
Positive integers:- 1, 2, 3,………
Negative integers:- -1, -2, -3, ………….
Non-negative integers:- 0, 1, 2, 3, ….

Some other numbers type:-


Even Number:- A counting number which divisible by 2 is called an even number.
Eg.:- 0, 2, 4, 6, 8, ……

Odd number:- A counting number which is not divisible by 2 is called an odd number.
Eg.:- 1, 3, 5, 7, ……….

Prime number:- A number is called a prime number if it has exactly two factors, namely itself and 1.
Eg.:- 2, 3, 5, 7, 11, 13, 17, 19, ……..

Co-primes number:- Two natural numbers a and b are said to be a co-prime if their HCF is 1.
Eg.:- (2, 3), (4, 5), (7, 9), (8, 11)… etc.

First post

Today is my first post on this blog. This blog is very useful for that person who is preparing for the competitive examination for any purpose like job or admission in any institution.

This blog contents both Arithmetic aptitude and computer technical problems.

Today we start with the important of the Aptitude. Aptitude is more important to our education life. These days every ware a small aptitude test to get the admission. It is not fully based upon the pure mathematics, it is actually a technical mathematic in which we solve the problem by using small tips and tricks.

Another subject of this blog is computer technical problem, which is useful to the computer engineers. It is related to the technical test which is organized by the industry at placement or campus time.

From the next post you get the information about aptitude and computer technical.

Thank you….