**Campus Recruitment – Quantitative Aptitude – Number System**

Contents

**Concepts**

In Hindu-Arabic system we use ten symbols 0,1, 2, 3, 4, 5, 6, 7, 8, 9 called digits to represent any number. This is the decimal system where we use the digits 0 to 9. Here 0 is called insignificant digit where as 1,……. ,9 are called significant digits.

**Classification of Numbers:**

**Natural Numbers:** The numbers 1, 2, 3, 4, 5, 6, ……which we use in counting are known as natural numbers. The set of all natural numbers can be represented by N = {1,2,3,4,5,…. }

**Whole Numbers:** If we include 0 among the natural numbers then the numbers 0, 1, 2, 3, 4, 5,…. are called whole numbers. Hence, every natural number is a whole number. The set of whole numbers is represented by W.

**Integers:** All counting numbers and their negatives including zero are known as integers.

The set of integers can be represented by Z or I.

Z = {…… -4,-3,-2,-1,0,1,2,3,4,…. }

Every natural number is an integer but every integer is not natural number.

**Positive Integers: **The set I + = {1, 2, 3,4, ….} is the set of all positive integers. Positive integers and Natural numbers are synonyms.

**Negative Integers:** The set I – = {…, -3, -2, -1} is the set of all negative integers. 0 (zero) is neither positive nor negative.

**Non Negative Integers:** The set {0,1, 2, 3,…} is the set of all non negative integers.

**Rational Numbers:** The numbers of the form p/q,where p and q are integers, p is not divisible by q and q≠ 0, are known as rational numbers.

**(or)** Any number that can be written in fraction form is a rational number. This includes integers, terminating decimals, and repeating decimals as well as fractions.

**e.g:** 3/7, 5/2, -5/9, 1/2, -3/5 etc

The set of rational numbers is denoted by Q.

**Irrational Numbers:** Any real number that cannot be written in fraction form is an irrational number. Numbers which are both, non-terminating as well as non-repeating decimals are called irrational numbers.

**e.g.:** Absolute value of 10/3,22/7,√2,√3,√10…

**Note**: A terminating decimal will have a finite number of digits after the decimal point.

**e.g.: **3/4 -=0.75,5/4-=1.25,25/16=1.5625.

**Repeating Decimals: **A decimal number that has digits that repeat forever.

**e.g.:** 1/3=0.333…. (here, 3 repeats forever.)

**Non-Repeating Decimal:** A decimal that neither terminates nor repeats.

**e.g.:√**2=1.4142135623….

**Real Numbers:** The rational and irrational numbers together are called real numbers.

**e.g.:**13/21, 2/5, -3/7, 4/2 etc are real numbers.

The set of real numbers is denoted by R.

**Even Numbers:** Any integer that can be divided exactly by 2.

**e.g.:** 2, 6, 0, -8, -10,….. are even numbers.

**Odd Numbers:** An integer that cannot be divided exactly by 2 is an Odd number.

**e.g.:** 1,3, -5, -7,…. are odd numbers.

**Prime Numbers:** A Prime Number can be divided evenly only by 1, or itself. And it must be a whole number greater than 1.

**e.g.:** Numbers 2,3,5, 7,11,13,17,…… are prime.

All primes which are greater than 3 are of the form (6n+l) or (6n-l).

**Note:**

• 1 is not a prime number.

• 2 is the least and only even prime number.

• 3 is the least odd prime number.

• Prime numbers up to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43,47, 53, 59, 61, 67, 71, 73, 79, 83,89,97. There are 25 prime numbers up to 100.

**Composite Number:** Natural numbers greater than 1 which are not prime, are known as composite numbers. The number 1 is neither prime nor composite.

Two numbers which have only 1 as the common factor are called co-primes (or) relatively prime to each other,

**e.g.:** 3 and 5 are co primes.

**Note:**

Natural Numbers = 1 + Prime + Composite Numbers.

Whole Numbers = 0 (Zero) + Natural Numbers.

Integers = Negative Integers + 0 + Positive Integers.

Real Numbers = Rational + Irrational Numbers.

**Test of Divisibility:**

Divisibility by 2: A number is divisible by 2 if the unit’s digit is either zero or divisible by 2.

e.g.: Units digit of 76 is 6 which is divisible by 2 hence 76 is divisible by 2.

Units digit of 330 is 0 so it is divisible by 2.

Divisibility by 3: A number is divisible by 3 if sum of all digits in it is divisible by 3.

e.g.: The number 273 is divisible by 3 since 2 + 7 + 3 = 12 which is divisible by 3.

Divisibility by 4: A number is divisible by 4, if the number formed by the last two digits in it is divisible by 4, or last two digits are zeros.

e.g.: The number 5004 is divisible by 4 since last two digits 04 is divisible by 4.

Divisibility by 5: A number is divisible by 5 if the units digit in the number is either 0 or 5.

e.g.: 375 is divisible by 5 as 5 is in the units place.

Divisibility by 6: A number is divisible by 6 if it is even and sum of all digits is divisible by 3.

e.g.: The number 6492 is divisible by 6 as it is even and sum of its digits 6 + 4 + 9 + 2 = 21 is divisible by 3.

Divisibility by 7:

Step-1: Remove unit’s digit. And double it.

Step-2: Subtract it from the rest of the number.

Step-3: Check whether the resulted number is divisible by 7 or not.

Step-4: Repeat the above steps until the resulted number is either 0 (zero) or divisible by 7. e.g.: Consider the number 10717.

Step-1: Removing the unit’s digit i.e. 7. Double of 7 =14. Step-2:1071 – 14 = 1057.

Step-3: Now remove 7 from 1057 and double it i.e. 14. Step-4:105-14 = 91.

Step-5: Now remove 1 and double it i.e. 2.

Step-6: 9-2 = 7

The final result 7 is divisible by 7. So the given number i.e. 10717 is also divisible by 7.

Divisibility by 8: A number is divisible by 8, if the number formed by last 3 digits is divisible by 8.

e.g.: The number 6573392 is divisible by 8 as the last 3 digits ‘392’ is divisible by 8.

Divisibility by 9: A number is divisible by 9 if the sum of its digit is divisible by 9.

e.g.: The number 15606 is divisible by 9 as the sum of the digits l+5 + 6 + 0 + 6 = 18is divisible by 9.

Divisibility by 10: Last digit should be zero,

e.g.: The last digit of 4470 is zero. So, it is divisible by 10.

Divisibility by 11: A number is divisible by 11 if the difference of the sum of the digits at odd places and sum of the digits at the even places is either zero or divisible by 11. (or) Subtract the first digit from a number made by the other digits. If that number is divisible by 11 then the original number is also divisible by 11.

e.g.: In the number 9823, the sum of the digits at odd places is 9+2=11 and the sum of the digits at even places is 8+3=11. The difference between them is 11 – 11 = 0. Hence, the given number is divisible by 11.

e.g.: 14641

1464 – 1 is 1463

146 – 3 is 143

14 – 3 = 11, which is divisible by 11, so 14641 is also divisible by 11.

If a number N is divisible by two numbers ‘a’ and ‘b’ where a,b are co primes then N is divisible by ab.

** Co-prime Numbers: **Two numbers are co-prime to each other if they have ‘no common factor except V.

Divisibility by 12: A number is divisible by 12 if it is divisible by 3 and 4.

e.g.: The number 1644 is divisible by 12 as it is divisible by 3 and 4. Here 3 and 4 because they are co-prime to each other.

Divisibility by 13: Iteratively add 4 times the last digit to the rest until you get a number divisible by 13 .

e.g.: 7462 => 746 + (2 x 4) = 754 => 75+ (4 x 4) = 91 91 is divisible by 13. So, 7462 is also divisible by 13. Divisibility by 14: The number is divisible by 7 and 2. Divisibility by 15: The number is divisible by 3 and 5.

Divisibility by 16:With a 3 digit number: Multiply hundreds digit by 4,

then add the last two digits.

e.g.: 352 => (3×4)+52 = 12 + 52 = 64

64 is divisible by 16. So, 352 is also divisible by 16.

With a more than 3 digit number: The last four digits form a number is divisible by 16. e.g.: 38512 => Here is 8512 is divisible by 16. So, 38512 is also divisible by 16.

Divisibility by 17:

Subtract 5 times the last digit from the rest.

e.g.: 3961 => 396 – (1×5) = 391 => 39 – (1×5) = 34

34 is divisible by 17. So, 3961 is also divisible by 17.

Divisibility by 18: An even number satisfying the divisibility test by 9 is also divisible by 18.

e.g.: The number 80388 is divisible by 18 as it satisfies the divisibility test of 9.

Divisibility by 19: Add twice the last digit to the rest,

e.g.: 10944 => 1094 + (4 x 2) = 1102

=> 110 + (2×2) = 114 => 11 + (4 x 2) = 11 + 8 = 19.

Divisibility by 20: Last digit is zero & tens digit is even, i e.g.: 980; Last digit is zero. And tens digit is even.

Divisibility by 25: A number is divisible by 25 if the number formed by the last two digits is divisible by 25 or the last two digits are zero.

e.g.: The number 7975 is divisible by 25 as the last two digits are divisible by 25.

**Common Factors:**

A common factor of two or more numbers is a number which divides each of them exactly.

e.g.: 3 is a common factor of 6 and 15.

**Highest Common Factor (HCF):**

Highest common factor of two or more numbers is the greatest number that divides each of them exactly.

e.g.: 3, 4, 6,12 are the factors of 12 and 36. Among them the greatest is 12. Hence the HCF of 12, 36 is 12.

HCF is also called as Greatest common divisor (GCD) or Greatest Common measure (GCM).

**Method of Finding HCF: Method of division**

**HCF of Two Numbers:**

Step 1: Greater number is divided by the smaller number.

Step 2: Divisor of step-1 is divided by its remainder.

Step 3: Divisor of step-2 is divided by its remainder. This could be continued until the remainder is 0.

Then HCF = Divisor of the last step.

e.g.: Find the HCF of 96 and 348.

**Explanation:** Here the divisor of the last step is 12. So,HCF of 96 and 348 is 12.

**CF of More than Two Numbers:**

Step 1: Take any two numbers as your wish and find their HCF.

Step 2: Now find the HCF of third number and HCF obtained for the previous two numbers.

Step 3: Now find the HCF of fourth number and HCF obtained in the previous step. Continue the same process till the last number. The final HCF is concluded to be the HCF of all the given numbers,

6 is HCF of 120, 246. Now take 3rd number (i.e. 100) and HCF obtained in the above step (i.e. 6) and find HCF.

**HCF of Decimals:**

e.g.: Find the HCF of 3.2, 4.12,1.3, 7.

**Explanation**: First eliminate the influence of decimals by multiplying it either by 10 or 100 or 1000 etc. Here multiply the numbers with 100 to make all the numbers decimal free. i.e. 320, 412,130, 700.

Now, find the HCF of above numbers. We get it as 2. Did you remember we multiplied all the numbers by 100 to eliminate the influence of decimals. Hence, now we divide the answer Jay 100 to get HCF of the original numbers. The HCF is =2/100=0.02

**LCM (Least Common Multiple):**

Least common multiple of two or more given numbers is the ‘least or lowest number’ which is divisible by each of them exactly. In the sense without a non zero remainder.

**Method of Finding LCM:**

Step-1: Write numbers in a line separated by comma.

Step 2: Divide any two of the given numbers exactly with a least possible prime number then the quotients and the undivided numbers are written in the next line.

Step 3: Repeat the same process till all the numbers in the line are prime to each other i.e. no more common factors exist.

**Key Points on LCM and HCF:**

1) HCF of fractions is always a fraction but LCM of fractions may be a fraction or an integer.

2) The product of any two numbers is equal to product of their LCM and HCF.

e.g.: What is LCM and HCF of 32 and 450 ?

a) 7200, 8 b) 7100, 2 c) 7800,2 d) 7200, 2

**Explanation**: Product of 32 and 450 is 14400

The LCM of 32 and 450 is 7200.

The HCF of 32 and 450 is 2.

(or) You can verify from options.

Option-a: 7200 x 8 ≠ 32 x 450.

Option-b: 7100 x 2 ≠ 32 x 450.

Option-c: 7800 x 2 ≠ 32 x 450.

Option-d: 7200 x 2 = 32 x 450.

3) To find the greatest number that will exactly divide x, y and z; Required number = HCF of x, y, z.

4) To find the greatest number that will divide x, y and z leaving remainders a, b and c respectively.

Required number = HCF of (x-a), (y-b) and (z-c).

5) To find the least number which is exactly divisible by x, y and z. Required number = LCM of x, y and z.

6) To find the least number which when divided by x, y, z leaves the remainders a, b, c respectively.

Then it is always observed that,

(x – a) = (y – b) = (z – c) = K (Assume).

Required number = (LCM of x, y and z) – K.

7) To find the least number which when divided by x, y and z leaves the same remainder r in each case. Required number = (LCM of x, y and z) + r.

8) To find the greatest number that will divide x, y and z leaving the same remainder in each case,

If the value of remainder r is given, then

Required number = HCF of (x-r), (y-r) and (z-r).

If the value of remainder is pot given, then

Required number = HCF of Ι (x-y) I, I (y-z) I, I (z-x) I.

**Complete Remainder:**

A remainder obtained by dividing a given number by the method of successive division is called complete remainder.

e.g.: A certain number when successively divided by 2, 3 and 5 leave remainders 1, 2 and 4 respectively. What is the complete remainder or remainder when the same number be divided by 30?

**Explanation**: For example, if a number when divided by 2, leaves remainder 1 would be of form = 2n + 1.

And a number when divided by 3, leaves remainder 2 would be of form = 3n + 2.

So, a number when successfully divided by 2,3, 5 leaves remainder 1, 2,4 would be of the form = 2[3(5n+4)+2]+l = 30n+29.

When (30n + 29) is divisible by 30, the remainder is 29.

1) When there are two divisors d1,d2 and two remainders r1, r2 the complete remainder is given by d1 r2+ r1.

2) When there are three divisors d1, d2, d3 and three remainders r1, r2, r3 the complete remainder is given by d1 d2 r3+ d1 r2+ r1.

3) When there are four divisors d1, d2, d3, d4 and four remainders r1, r2, r3, r4 the complete remainder is given by d1 d2 d3 r4+d1 d2 r3+d1 r2+r1.

4) In any case if there are no remainders consider them as zeros.

**Fractions and Ordering Fractions:**

1) In the fraction 3/4; bottom number (denominator) says how many parts the whole is divided into. The top number (the numerator) says how many parts we have.

2) Fraction =Numerator/Denominator

Such a fraction is known as common fraction.

3) A fraction whose denominator is 10 or 100 or 1000 etc is called a decimal fraction.

4) Fractions whose denominators are same are called like fractions. For example,3/7,5/7.

5) Fractions whose denominators are different are called unlike fractions. For example, 3/4,5/13.

6) When two fractions have the same denominator, the greater fraction is that which has greater numerator.

7) When two fractions have the same numerator, the greater fraction is that which has the smaller denominator.

8) If the identity is not possible, convert the fraction into the convenient form.

e.g:3/5, 13/16, 5/7, 97/104 in ascending order.

**Explanation**: LCM of 5,16, 7,104 is 7280.

Now multiply the numerator and denominator of the fractions with a number such that the denominator equals 7280.

Let us observe the above working rule in words.

Step 1: Group the digits in pairs, starting with the digit in the units place.

Step 2: Think of the largest number whose square is equal to or just less than the first pair. Take this number as the divisor and also as the quotient.

In the given example, largest number whose square is near to 6 is 2 (i.e. 2^{2} = 4). So, 2 is the divisor and quotient.

Step 3: Subtract the product of the divisor and the quotient from the first pair and bring down the next pair to the right of the remainder. This becomes the new dividend.

Step 4: Double the quotient and put a blank for a number beside it (i.e. 4[?]). Now think of a largest number (for example 5) to fill in the blank in such a way that the product of a new divisor (i.e. 45) and this digit (i.e. 5) is equal to or less than new dividend (i.e. 245).

Step 5: Repeat steps (2), (3) and (4) till all the pairs have been taken up. Now, the quotient so obtained is the required square root of the given number.

**Conceptual Examples**

**Exercise**

**Explanations**

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