Quantitative Aptitude Important Formulas

H.C.F and L.C.M - Important Formulas,Tricks and Examples

Factors and Multiples: If a number x divides another number y exactly, we say that x is a factor of y. Also y is called a multiple of x.

Highest Common Factor (HCF)
The H.C.F. of two or more than two numbers is the greatest number that divides each one of them exactly. There are two methods for determining H.C.F.:

Prime factorization method
Step 1: Express each number as a product of prime factors.
Step 2: HCF is the product of all common prime factors using the least power of each common prime factor

Division Method
Step 1: Write the given numbers in a horizontal line separated by commas.
Step 2: Divide the given numbers by the smallest prime number (write in the left side) which can exactly divide all the given numbers.
Step 3: Write the quotients in a line below the first.
Step 4: Repeat the process until we reach a stage where no common prime factor exists for all the numbers.
Step 5: We can see that the factors mentioned in the left side clearly divides all the numbers exactly and they are common prime factors. Their product is the HCF


Lowest Common Multiple (LCM)
The L.C.M. of two or more than two numbers is the least number which is exactly divisible by each one of the given numbers.

Prime factorization method
Step 1 : Express each number as a product of prime factors.
Step 2 : LCM is The product of highest powers of all prime factors.

Division Method
Step 1: Write the given numbers in a horizontal line separated by commas.
Step 2: Divide the given numbers by the smallest prime number which can exactly divide at least two of the given numbers.
Step 3: Write the quotients and undivided numbers in a line below the first.
Step 4: Repeat the process until we reach a stage where no prime factor is common to any two numbers in the row.
Step 5: LCM is The product of all the divisors and the numbers in the last line.

HCF and LCM of Fractions
LCM of fractions =LCM of Numerators/HCF of Denominators
HCF of fractions =HCF of Numerators/LCM of Denominators

Note:
Product of two numbers = Product of their HCF and LCM.

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Numbers - Important Formulas,Tricks and Examples

The common number system is decimal number system.
The Decimal number system used base (or radix) as 10 . This means that the system has ten symbols or numerals to represent any quantity. These symbols are called Digits and they are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Types of Numbers

Natural Numbers:The numbers that are used for counting called Natural Numbers.These are infinite and start from the number 1 .
Ex : 1, 2, 3, 4, 5, 6 ........

Whole numbers:The whole numbers are just all the natural numbers plus zero.
Ex : 0, 1, 2, 3, 4, 5 .............

Integers: Integers incorporate all positive and negative numbers with zero.
Ex : .... –3, –2, –1, 0, 1, 2, 3 .....

Even Numbers: An even number is one that can be divided evenly by two leaving no remainder, such as 2, 4, 6, and 8.

Odd Numbers: An odd number is one that does not divide evenly by two, such as 1, 3, 5, and 7.

Rational Numbers: All numbers of the form p/q where p and q are integers (q ≠ 0) called Rational numbers.
Ex : 4, 3/4, 0, ….

Irrational Numbers: Irrational numbers are the opposite of rational numbers. An irrational number cannot be written as a fraction as p/q where a and b are integers.
the decimal values for irrational numbers never end and do not have a repeating pattern in them.please note ‘pi’ has never ending decimal places, is irrational.
Ex : pi, 2^1/2 , 3^1/2, 5^1/2 , 7^1/2 ..........

Real numbers: Real numbers include counting numbers, whole numbers, integers, rational numbers and irrational numbers.
Ex : 8, 6, 2, (3)^1/2 , 3/5 etc.

Prime Number: A prime number is a number which can be divided only by 1 and itself. The prime number has only two factors, 1 and itself.
Ex : 2, 3, 7, 11, 13, 17, …. are prime numbers.

Composite Number: A Composite Number is a number which can be divided evenly. Any composite number has additional factors than 1 and itself.
Ex : 4, 6, 8, 9, 10 …..

Co-primes or Relatively prime numbers: A pair of numbers not having any common factors other than 1 or –1. (Or alternatively their greatest common factor is 1 or –1)
Ex : 15 and 28 are co-prime, because the factors of 15 (1,3,5,15) , and the factors of 28 (1,2,4,7,14,28) are not in common (except for 1) .

Twin Primes: A pair of prime numbers is said to be twin primes if they differ by 2.
Ex : (3,5) , (5,7) , (11,13) , …

Algebra

(a+b)2 = a2 + 2ab + b2
(a-b)2 = a2 - 2ab + b2

(a+b)2 - (a-b)2 = 4ab
(a+b)2 + (a-b)2 = 2 (a2 + b2)

a2-b2 = (a-b)(a+b)

(a+b)3 = a3+b3+3ab(a+b) = a3+b3+3a2b+3ab2
(a-b)3 = a3-b3-3ab(a-b) = a3-b3-3a2b+3ab2

a3+b3 =(a+b)3-3ab(a+b) = (a+b)(a2-ab+b2)
a3-b3 =(a-b)3+3ab(a-b) = (a-b)(a2+ab+b2)

Test of Divisibility

Divisibility By 2 : A number is divisible by 2, if its unit's digit is zero or even (2, 4, 6, 8..).

Divisibility By 3 : A number is divisible by 3, if the sum of its digits is divisible by 3.

Divisibility By 4 : A number is divisible by 4, if the number formed by the last two digits is divisible by 4.

Divisibility By 5 : A number is divisible by 5, if its unit's digit is either 0 or 5.

Divisibility By 6 : A number is divisible by 6, if it is divisible by both 2 and 3.

Divisibility By 8 : A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.

Divisibility By 9 : A number is divisible by 9, if the sum of its digits is divisible
by 9.

Divisibility By 10 : A number is divisible by 10, if it ends with 0.

Divisibility By 11 : A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.

Divisibility By 12: A number is divisible By 12 if the number is divisible By both 4 and 3.

Divisibility By 13: A number is divisible By 13 if its unit’s place digit is multiplied By 4 and added to the remaining digits and the number obtained is divisible By 13.

Divisibility By 14: A number is divisible By 14 if the number is divisible By both 2 and 7.

Divisibility By 15: A number is divisible By 15 if the number is divisible By both 3 and 5.

Divisibility By 16: A number is divisible By 16 if its last 4 digits is divisible By 16 or if the last four digits are zeros.

Divisibility By 17: A number is divisible By 17 if its unit’s place digit is multiplied By 5 and subtracted from the remaining digits and the number obtained is divisible By 17.

Divisibility By 18: A number is divisible By 18 if the number is divisible By both 2 and 9.

Divisibility By 19: A number is divisible By 19 if its unit’s place digit is multiplied By 2 and added to the remaining digits and the number obtained is divisible By 19.

Divisibility By 20: A number is divisible by 20 if it is divisible by 10 and the tens digit is even.

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Pipes and Cistern

Three taps A,B and C can fill a tank in 12, 15 and 20 hours respectively. If A is open all the time and B and C are open for one hour each alternatively, the tank will be full in?

Two pipes A and B can fill a tank in 6 hours and 4 hours respectively. If they are opened on alternate hours and if pipe A s opened first, in how many hours, the tank shall be full?

A large tanker can be filled by two pipes A and B in 60 min and 40 min respectively. How many minutes will it take to fill the tanker from empty state if B is used for first half of the time, and A and B fill it together for the other half?

A tank is filled by 3 pipes with uniform flow. The first two pipes operating simultaneously fill the tank in the same time during which the tank is filled by the third pipe alone. The 2nd pipe fills the tank 5 hours faster than first pipe and 4 hours slower than third pipe. The time required by first pipe is?

Two pipes A and B together can fill a cistern in 4 hours. Had they been opened separately, then B would have taken 6 hours more than A to fill the cistern. How much time will be taken by A to fill the cistern separately?

Two pipes A and B can fill a tank in 24 min and 32 min respectively. If both the pipes are opened simultaneously, after how much time B should be closed so that the tank is full in 18 min?

Two pipes A and B can fill a tank in 36 min and 45 min respectively. A water pipe C can empty the tank in 30 min. First A and B are opened. after 7 min, C is also opened. In how much time, the tank is full?

Two pipes can fill a cistern in 14 hours and 16 hours respectively. The pipes are opened simultaneously and it is found that due to leakage in the bottom it took 32 min more to fill the cistern. When the cistern is full, in what time will the leak empty it?

Bucket P has thrice the capacity as Bucket Q. It takes 60 turns for Bucket P to fill the empty drum. How many turns it will take for both the buckets P and Q, having each turn together to fill the empty drum?

Probability

In a class , 30 % of the students offered English, 20 % offered Hindi and 10 % offered Both.If a student is offered at random, what is the probability that he has offered English or Hindi?

In a lottery ,there are 10 prizes and 25 blanks.A lottery is drawn at random. what is the probability of getting a prize ?

Two dice are thrown simultaneously .what is the probability of getting two numbers whose product is even?

Two diced are tossed the probability that the total score is a prime number?

Two cards are drawn at random from a pack of 52 cards What is the probability that either both are black or both are queens?

Two dice are thrown together .What is the probability that the sum of the number on the two faces is divisible by 4 or 6?

A bag contains 6 white and 4 black balls .Two balls are drawn at random .Find the probability that they are of the same colour?

In a simultaneous throw of a pair of dice,find the probability of getting a total more than 7?

An unbiased die is tossed.Find the probability of getting a multiple of 3?

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