Indices and Surds

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Indices :-
Indices are used to describe the general term for 2 in say x^2. There are a few laws to know when manipulating expressions involving indices.
























Surds :-
Surds are basically an expression involving a root, squared or cubed etc...
There are some basic rules when dealing with surds.


Also notice the special case




Difference of Two Squares


This is called the difference of two squares .

Rationalising Surds:
When you have a fraction where both the nominator and denominator are surds, rationalising the surd is the process of getting rid of the surd on the denominator. To rationalise a surd you multiply top and bottom by fraction that equals one. Take the example shown below
1/√2
To rationalise this multiply by effectively 1
1/√2 * √2 /√2
Can you see why √2 /√2 was chosen? This is because √2 * √2 = 2 so the denominator becomes surd free.

For a more complex term

HCF and LCM

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Prime factorisation
A prime factorisation of a natural number can be expressed in the exponential form.
For example:(i) 48 = 2 x 2 x 2x 2 x3 = 24 x 3(ii) 420 = 2 x 2 x 3 x 5 x 7 = 22 x 3 x 5 x 7

Least Common Multiple (L.C.M.):
A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, ....

The least common multiple (LCM) of two numbers is the smallest number (not zero) that is a multiple of both.

Method 1:
Simply list the multiples of each number (multiply by 2, 3, 4, etc.) then look for the smallest number that appears in each list.
Example: Find the least common multiple for 5, 6, and 15.
Multiples of 5 are 10, 15, 20, 25, 30, 35, 40,...
Multiples of 6 are 12, 18, 24, 30, 36, 42, 48,...
Multiples of 15 are 30, 45, 60, 75, 90,....
Now, when you look at the list of multiples, you can see that 30 is the smallest number that appears in each list.Therefore, the least common multiple of 5, 6 and 15 is 30.

Method 2
To use this method factor each of the numbers into primes. Then for each different prime number in all of the factorizations, do the following...
1. Count the number of times each prime number appears in each of the factorizations.
2. For each prime number, take the largest of these counts.
3. Write down that prime number as many times as you counted for it in step 2.
The least common multiple is the product of all the prime numbers written down.

Example: Find the least common multiple of 5, 6 and 15.
Factor into primes
Prime factorization of 5 is 5
Prime factorization of 6 is 2 x 3
Prime factorization of 15 is 3 x 5
· Notice that the different primes are 2, 3 and 5.
Now, we do
Step #1 - Count the number of times each prime number appears in each of the factorizations...
The count of primes in 5 is one 5
The count of primes in 6 is one 2 and one 3
The count of primes in 15 is one 3 and one 5

Step #2 - For each prime number, take the largest of these counts. So we have...
The largest count of 2s is one
The largest count of 3s is one
The largest count of 5s is one

Step #3 - Since we now know the count of each prime number, you simply - write down that prime number as many times as you counted for it in step 2.
Here they are...2, 3, 5

Step #4 - The least common multiple is the product of all the prime numbers written down.
2 x 3 x 5 = 30 Therefore, the least common multiple of 5, 6 and 15 is 30.
So there you have it. A quick and easy method for finding least common multiples.

Highest Common Factor (abbreviated H.C.F.) :
HCF of two natural numbers is the largest common factor (or divisor) of the given natural numbers. In other words, H.C.F. is the greatest element of the set of common factors of the given numbers.

H.C.F. is also called Greatest Common Divisor (abbreviated G.C.D.)

Example. Find the H.C.F. of 72, 126 and 270.
Solution. Using Prime factorisation
72 = 2 x 2 x 2 x 3 x 3 = 2 ^3 x 3 ^2
126 = 2 x 3 x 3 x 7 = 2 ^ 1 x 3 ^2 x 7 ^1
270 = 2 x 3 x 3 x 3 x 5 =2 ^1 x 3 ^3 x 5 ^1

H.C.F. of the given numbers = the product of common factors with least index= 2 ^1 x 3 ^2

Using Division method
First find H.C.F. of 72 and 126
72\126\1
72
54\ 72\1
54
18\ 54 \3
54
0
H.C.F. of 72 and 126 = 18Similarly calculate H.C.F. of 18 and 270 as 18Hence H.C.F. of the given three numbers = 18

Co-prime numbers : Two natural numbers are called co-prime numbers if they have no common factor other than 1.
in other words, two natural numbers are co-prime if their H.C.F. is 1.

Some examples of co-prime numbers are: 4, 9; 8, 21; 27, 50.

Relation between L.C.M. and H.C.F. of two natural numbers

The product of L.C.M. and H.C.F. of two natural numbers = the product of the numbers.

Note. In particular, if Two natural numbers are co-prime then their L.C.M. = The product of the numbers.

Important

For LCM
1. LCM >= the largest number of a set.
2. For Co-prime numbers - LCM is product of co-prime numbers.
3. LCM is multiple of all the numbers of a set as well as HCF of a set of numbers.

For HCF
1. HCF =< the largest number of a set.
2. For Co-prime numbers - HCF is equals to 1.
3. HCF is factor of all the numbers of a set as well as LCM of a set of numbers.

Divisibility Rules

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Dividing by 2
All even numbers are divisible by 2. E.g., all numbers ending in 0,2,4,6 or 8.

Dividing by 3
Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 3, so is the number
For example: 12123 (1+2+1+2+3=9) 9 is divisible by 3, therefore 12123 is too!


Dividing by 4
Are the last two digits in your number divisible by 4?
If so, the number is too!
For example: 358912 ends in 12 which is divisible by 4, thus so is 358912.

Dividing by 5
Numbers ending in a 5 or a 0 are always divisible by 5.

Dividing by 6
If the Number is divisible by 2 and 3 it is divisible by 6 also.

Dividing by 7 (2 Tests)
Take the last digit in a number.
Double and subtract the last digit in your number from the rest of the digits.
Repeat the process for larger numbers.
Example: 357 (Double the 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7 and we can now say that 357 is divisible by 7.

NEXT TEST
Take the number and multiply each digit beginning on the right hand side (ones) by 1, 3, 2, 6, 4, 5. Repeat this sequence as necessary
Add the products.
If the sum is divisible by 7 - so is your number.
Example: Is 2016 divisible by 7?
6(1) + 1(3) + 0(2) + 2(6) = 21
21 is divisible by 7 and we can now say that 2016 is also divisible by 7.

Dividing by 8
This one's not as easy, if the last 3 digits are divisible by 8, so is the entire number.
Example: 6008 - The last 3 digits are divisible by 8, therefore, so is 6008.

Dividing by 9
Almost the same rule and dividing by 3. Add up all the digits in the number.
Find out what the sum is. If the sum is divisible by 9, so is the number.
For example: 43785 (4+3+7+8+5=27) 27 is divisible by 9, therefore 43785 is too!

Dividing by 10
If the number ends in a 0, it is divisible by 10.

Dividing by 11
The difference between the sum of the odd numbered digits (1st, 3rd, 5th...) and the sum of the even numbered digits (2nd, 4th...) is divisible by 11.

Divisibility by 11 is the most interesting of the above tests . We do two sums (the odd numbered digits and the even numbered digits), subtract one sum from the other, and see if this is divisible by 11. By the way, if we end up with zero, then that is divisible by 11. We can repeat that process, just as we did with 3. Let's look at an example:

34871903
3+8+1+0=12
4+7+9+3=23
23-12=11 Is divisible by 11
We can, of course, do the summing in different orders. In fact we can just go from left to right adding and subtracting alternate digits: 3-4+8-7+1-9+0-3=-11 (divisible by 11).

Note: For the time being, I will not deal with divisibility by larger numbers. Since they are hardly useful in exams like CAT, GMAT, GRE, SNAP and etc.

Converting Recurring Numbers in Fractions

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Converting a recurring decimal to vulgar fraction A decimal with recurring value is called recurring decimal.
E.g: 2/9 will give 0.22222222...... where 2 is recurring number.

Method:
a) Separate the recurring number from the decimal fraction.
b) Annex denominator with "9" as many times as the length of the recurring number.
c) Reduce the fraction to its lowest terms.

Example: Consider 0.2323232323

Step a: The recurring number is 23
Step b: 23/99 [the number 23 is of length 2 so we have added two nines]
Step c: Reducing it to lowest terms : 23/99 [it can not be reduced further].

How to Convert a mixed-recurring decimal to vulgar fraction ? A decimal with both recurring and non-recurring value is called mixed recurring decimal.
E.g: 28/25 will give 1.1199999999...... where 11 is non-recurring number and 9 is recurring number.

Method:
a) Separate the recurring number, non recurring number from the decimal fraction.
b) Round the decimal after point to the first recurring value.
c) Result of step b - non recurring number.
d) Annex as many "0" as non-recurring number length and as many "9" as recurring number length.
e) Step c / Step d
f) Add the fraction with the number before decimal point.
Example: Consider 1.11999999...

Step a: The recurring number is 11, non-recurring number is 9
Step b and c: 119-11 [rounded value of number after decimal point - non recurring value]
Step d: 900
Step e: 108/900 [c/d]
Step f: 1+¹⁰⁸/₉₀₀ [adding with number before decimal point ]
Reducing it to lowest terms : 900+108 / 900 = 1008/900 = 28/25.

This is a very important topic as time is concern.

Basic Classification

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Number: - A number is a mathematical object used in counting and measuring. A notational symbol which represents a number is called a numeral.

Real Number: - The real numbers include all of the measuring numbers.

Imaginary Number: - Number with square root of negative number is called imaginary number. These numbers are in the form of a + ib.

Rational Number: - A rational number is a number that can be expressed as a fraction with an integer numerator and a non-zero natural number denominator.

Irrational Number: - In mathematics, an irrational number is any real number which cannot be expressed as a fraction m/n, where m and n are integers, with n non-zero and is therefore not a rational number.

Integers: - The integers are formed by the natural numbers including 0 (0, 1, 2, 3, ...) together with the negatives of the non-zero natural numbers (−1, −2, −3, ...). Viewed as subset of the real numbers, they are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2 ...}.

Fractions: - A fraction is a number that can represent part of a whole.

Proper Fraction: Value of the fraction is less than 1. These are in the form of ½, 2/3 , ¾ ..

Improper Fraction: - Value of the fraction is greater than 1. These are in the form of 3/2 , 4/3 …

Mixed Fraction: - When an integer is given before proper fraction then number is called mixed fraction. Numbers of the form 2¾.

Whole Number: All non negative numbers including Zero. W ~ (0,1,2,3,4...)

Natural Numbers: - All the number starting from 1. N ~ (1,2,3,4...)

Natural Numbers = Whole Number - Zero

Important Notes

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Remember:-

1. Last Digit of any [Even Number] ^ 4n = 6

Last Digit of any [Odd Number] ^ 4n = 1

There is exception to above rule. This rule is not applicable for number ending with 0, 1, 5 and 6 because cyclicity of these numbers is 1.

2. [25]^n always ends with 25.

[76]^n always end with 76.

3. Perfect Number: -

A number is a perfect number when sum of all its factors excluding number itself equals to number.

Eg. 6 Factors à 1, 2, 3 and 6

Ie 1 + 2 + 3 = 6

28, 496 and 8128 are few of such numbers.

4. Consider a number formed with digit a and b ie ab then number formed by reversing the digit with be ba then

ab = 10 a+ b

ba = 10 b + a

=> ab + ba = 11 (a + b) & ab – ba = 9 (a-b)

5. Product of three consecutive numbers is always divisible by 3. ie number of the form

(p-1)p(p+1) is always divisible by 3.

Number Cyclicity

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Number Cyclicity:-

Number cyclicity is very important for every chapter of the quant. Any exams will definitely need the basics of number cyclicity. Questions can directly be asked on this topic and you will need this to solve many questions on other topics.

Don’t forget time is biggest constrain for any entrance exam and cyclicity will have one to save time.

Cyclicity is basically use to find the unit digit or tens digit of the number.

Unit Digit Cyclicity:-

Q1. Find the unit digit of 2^2548.

Sol: - You will need more than 6 hours to solve this problem if you don’t use cyclicity theorem.

We notice that

2^1 end with 2

2^2 end with 4

2^3 end with 8

2^4 end with 6

2^5 end with 2

2^6 end with 4

2^7 end with 8

2^8 end with 6

We notice that 5^th power end in 2 and number repeats after 4 powers. Hence cyclicity for 2 is 4. It will always end with 2, 4, 6 and 8.

So Divide 2548 by 4 and we get remainder = 2

Hence unit digit of 2^2548 with be 4.

Remember:-

When Remainder is 1 number ends with 2

When Remainder is 2 number ends with 4

When Remainder is 3 number ends with 8

When Remainder is 0 or 4 number ends with 6.

Similarly we can find of all other numbers.

Tens Digit Cyclicity:-

Similarly we can arrive for tens digit cyclicity:-

Number Tens Digit Cyclicity

One needs to practice much to get use to in this topic. Very Soon I will put examples on this topic and link will be given below.

Prime Numbers and Co-Prime Numbers

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Prime Numbers:-
Number which does not have any factor other than 1 and number itself are called Prime Numbers.

Eg. 2, 3, 7, 11 ….

Large Prime Number can be calculated by the following method :

Method:
This method can be used to calculate primes larger then 100.

Procedure:
a) Calculate the whole number (w) greater then square root of n.
b) Find the prime numbers[p(x)] less than the whole number(w).
c) Now check if the given number n is divisible by any of the prime numbers p(x).
d) If it gets divided then the number n is not a prime number else it is a perfect prime number.

Example:
Validate if 131 is prime number.

Step1: Square root of 131 = 11.44
So rounding it to the nearest large whole number it becomes 12.

Step2: Prime Numbers less than 12 are 2,3,5,7,11

Step3:
Given number 131 is not divisible by 2 or 3 or 5 or 7 or 11
So the given number is a prime perfect number.

Composite Numbers:-
Numbers having more than 2 factors are known as composite numbers.

Eg. 4, 6, 8 …

Remember:-

1. 2 is only even prime number.
2. 4 is first composite number.
3. 1 is neither a prime nor a composite number.
4. Prime numbers are always in form of 6K+ 1 or 6K -1 but vice versa not true.

Remember:-
If x is a prime number then number of the form (a^x – a) is always divisible by x.

Remember:-
A composite number n which is not a perfect square has one factor greater than √ n one less than √ n , product of these factors will give the number.

Remember:-
If p is a prime number such that (p >3) the
p^2 = 6K + 1 or
p^2 = 24k + 1

One just needs to remember the above rule. Still for deriving that formula one can Google to get it.

Twin Prime Numbers:-
Pair of prime numbers with the difference of 2 is called twin prime number.

Eg. (3,5) , (5,7) , (11,13) and go on.

Co-Prime Numbers (Relatively Prime Numbers):-

Pair of numbers that do not have any common factor except 1 are known as Co-prime numbers.

Eg . (9,16)
Factors of 9 -> 1, 3, 9
Factors of 16 -> 1, 2, 4, 8, 16

Remember:-
1. One is co-prime with any other number. (1,2) , (1,3) …
2. Two Consecutive numbers are always co-prime.
3. A prime number is co-prime with all numbers except its multiple.
4. Two prime numbers are always co-prime.

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Basic Operations on Numbers

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Let a and b any two numbers then find the results of basic operations below.








Points to remember:-
1. Sum and Difference of any two odd or two even numbers is always even.
2. a^b always depends on a.
ie Even raise to any number is even and Odd raise to any number is odd.
eg 2^ 3 = 8 and 3^ 2 = 9.
3. If atleast one number is even - Product is always even.

Classification of Numbers

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It can further be classified as
1. Basic Classification (Link)
2. Basic Operations ( Link )
3. Prime Numbers and Co-Prime Numbers ( Link)
4. Converting Recurring Numbers in Fractions (Link)
5. Base System
6. Number Cyclicity (Link)
7. Important Notes (Link)