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Shortcuts, Formulas & Tips

**Vol. 1: Number System & Arithmetic
**

*present *

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**Glossary
**

**Natural Numbers:** 1, 2, 3, 4…..

**Whole Numbers:** 0, 1, 2, 3, 4…..

**Integers:** ….-2, -1, 0, 1, 2 …..

**Rational Numbers:** Any number which can be expressed
as a ratio of two integers for example a p/q format
where ‘p’ and ‘q’ are integers. Proper fraction will have
(p<q) and improper fraction will have (p>q)

**Factors:** A positive integer ‘f’ is said to be a factor of a
given positive integer 'n' if f divides n without leaving a
remainder. e.g. 1, 2, 3, 4, 6 and 12 are the factors of 12.

**Prime Numbers:** A prime number is a positive number
which has no factors besides itself and unity.

**Composite Numbers:** A composite number is a number
which has other factors besides itself and unity.

**Factorial:** For a natural number 'n', its factorial is
defined as: n! = 1 x 2 x 3 x 4 x .... x n (Note: 0! = 1)

**Absolute value:** Absolute value of x (written as |x|) is
the distance of 'x' from 0 on the number line. |x| is
always positive. *|x| = x for x > 0* OR *-x for x < 0*

**Tip: ** The product of ‘n’ consecutive natural
numbers is always divisible by n!

**Tip: ** Square of any natural number can be written
in the form of 3n or 3n+1. Also, square of any
natural number can be written in the form of 4n or
4n+1.

**Tip: ** Square of a natural number can only end in 0,
1, 4, 5, 6 or 9. Second last digit of a square of a
natural number is always even except when last
digit is 6. If the last digit is 5, second last digit has
to be 2.

**Tip: ** Any prime number greater than 3 can be
written as 6k ±1.

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**Laws of Indices
**

�������� × �������� = ��������+����

�������� ÷ �������� = ��������−����

(��������)���� = ������������

����� 1 ����� = √������

����−���� = 1 ��������

����� ���� ����� = √������������

����0 = 1

**Last digit of an
**

**n(Right)
a(Down) 1 2 3 4 Cyclicity
**

**0 **0 0 0 0 **1
1 **1 1 1 1 **1
2 **2 4 8 6 **4
3 **3 9 7 1 **4
4 **4 6 4 6 **2
5 **5 5 5 5 **1
6 **6 6 6 6 **1
7 **7 9 3 1 **4
8 **8 4 2 6 **4
9 **9 1 9 1 **2
**

**Tip: ** If am = an, then m = n

**Tip: ** If am = bm and m ≠ 0;
Then *a = b * if m is Odd
Or *a = *±* b* if m is Even

**Tip: ** The fifth power of any number has the same units
place digit as the number itself.

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**HCF and LCM
**For twonumbers, HCF x LCM = product of the two.

HCF of Fractions = ������ �������� ������������������������������������ ������ �������� ��������������������������������������������

LCM of Fractions = ������ �������� ������������������������������������ ������ �������� ��������������������������������������������

**Relatively Prime or Co-Prime Numbers:** Two positive
integers are said to be relatively prime to each other if
their highest common factor is 1.

**Factor Theory
**If N = xaybzc where x, y, z are prime factors. Then,

Number of factors of N = **P** = (a + 1)(b + 1)(c + 1)

Sum of factors of N = x a+1 – 1 ��−1

�� y b+1 – 1 ��−1

�� z c+1 – 1 ��−1

Number of ways N can be written as product of two factors = P/2 or (P+1)/2 if P is even or odd respectively

The number of ways in which a composite number can be resolved into two co-prime factors is 2m-1, where m is the number of different prime factors of the number.

Number of numbers which are less than N and co-prime to ∅(��) = �� �1− 1

�� � �1− 1

�� � �1− 1

�� � {Euler’s Totient}

**Tip: ** If a, b and c give remainders p, q and r
respectively, when divided by the same number H,
then H is HCF of (a-p), (b-q), (c-r)

**Tip: ** If the HCF of two numbers ‘a’ and ‘b’ is H, then,
the numbers (a+b) and (a-b) are also divisible by H.

**Tip: ** If a number N always leaves a remainder R when
divided by the numbers a, b and c, then N = LCM (or a
multiple of LCM) of a, b and c + R.

**Tip: ** If N = (2)a(y)b(z)c where x, y, z are prime factors
Number of even factors of N = (a)(b+1)(c+1)
Number of odd factors of N = (b+1)(c+1)

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**Divisibility Rules
**
A number is divisible by:
**2, 4 & 8** when the number formed by the last, last two,
last three digits are divisible by 2,4 & 8 respectively.
**3 & 9** when the sum of the digits of the number is
divisible by 3 & 9 respectively.
**11** when the difference between the sum of the digits in
the odd places and of those in even places is 0 or a
multiple of 11.
**6, 12 & 15** when it is divisible by 2 and 3, 3 and 4 & 3
and 5 respectively.
**7,** if the number of tens added to five times the number
of units is divisible by 7.
**13,** if the number of tens added to four times the
number of units is divisible by 13.
**19,** if the number of tens added to twice the number of
units is divisible by 19.

**Algebraic Formulae
**

**a3 ± b3 = (a ± b)(a2 **∓** ab + b2). **Hence, *a3 ± b3 is divisible
by (a ± b) and (a2 ± ab + b2).*

**an - bn = (a – b)(an-1 + an-2b+ an-3b2 + ... + bn-1**)[for all n].
Hence, *an - bn is divisible by a - b for all n*.

**an - bn = (a + b)(an-1 – an-2b + an-3b2 ... – bn-1)**[n-even]
Hence, *an - bn is divisible by a + b for even n.*

**an + bn = (a + b)(an-1 – an-2b + an-3b2 + ... + bn-1)**[n-odd]
Hence, *an + bn is divisible by a + b for odd n.*

**a3 + b3 + c3 - 3abc = (a + b + c)(a2 + b2 + c2 - ab - ac - bc)
**Hence, *a3 + b3 + c3 = 3abc if a + b + c = 0
*

For ex., check divisibility of 312 by 7, 13 & 19

**For 7:** 31 + 2 x 5 = 31 + 10 = 41 ** Not divisible**

**For 13:**31 + 2 x 4 = 31 + 8 = 39

**. **

*Divisible***For 19:**31 + 2 x 2 = 31 + 4 = 35

**.**

*Not divisible*© 2013 www.oliveboard.com. All Rights Reserved

**Remainder / Modular Arithmetic
**

������� �� ∗ �� ∗ ��… ��

� = ������� �� �� � ∗ �������

�� �� � ∗ �������

�� �� �…

������� ��+ ��+ ��…

�� � = �������

�� �� � + �������

�� �� �+ �������

�� �� �…

**Case 1** – When the dividend (M) and divisor (N) have a
factor in common (k)

��������� �� � = �����������

���� � = �� ���������

�� �

*Example: ������ *�3
15

15 � = 3 ������ �3

14

5 � = 3 ∗ 4 = 12

**Case 2** – When the divisor can be broken down into
smaller co-prime factors.

������ ��� ���� � = ������ � ��

����∗����
� *{HCF (a,b) = 1}*

Let ������ ��� ���� � = ��1 & ������ �

�� ���� � = ��2

��������� �� � = ���������� + ������������ {Such that ax+by = 1}

*Example: ������ *�7
15

15 � = ������ �7

15

3∗5 �

������ �7 15

3 � = 1 & ������ �7

15

5 � = ������ �2

15

5 � = 3

������ �7 15

15 � = 3 ∗ ���� ∗ 3 + 5 ∗ �� ∗ 1

*{Such that 3x+5y=1}
*

Valid values are x = -3 and y = 2

������ �7 15

15 � = 9���� + 5�� = −17 ≡ 13

**Case 3 – Remainder when ��(����) **= ������������ + ������������−1 +
����������−2… is divided by (���� − ����) the remainder is ��(����)

**Tip: **If f(a) = 0, (x-a) is a factor of f(x)

*Continued >> *

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**Remainder Related Theorems
**

**Euler’s Theorem**:

Number of numbers which are less than N = �������� ∗ �������� ∗ ������ and co-prime to it are

∅(��) = �� �1− 1 ���� � �1− 1

���� � �1− 1

���� �

If M and N are co-prime ie HCF(M,N) = 1

��������� ∅(��)

�� � = ����

*Example: ������ *�7
50

90 � =?

∅(90) = 90 �1− 1 2 � �1− 1

3 � �1− 1

5 �

∅(90) = 90 ∗ 1 2 ∗ 2 3 ∗ 4 5

= 24

������ �7 24

90 � = 1 = ������ �7

48

90 �

������ �7 50

90 � = ������ �7

2

90 � ∗ ������ �7

48

90 � = 49 ∗ 1 = 49

**Fermat’s Theorem:
***If N is a prime number* and *M and N are co-primes*

��������� ��

�� � =��

��������� ��−����

�� � = ����

*Example: ������ *�6
31

31 � = 6 & ������ �6

30

31 � = 1

**
Wilson’s Theorem
***If N is a prime number*
�������(��−����)!

�� � = ��− ����

�������(��−��)! �� � = ����

*Example: ������ *�30!
31
� = 30 & ������ �29!

31 � = 1

**Tip: ** Any single digit number written (P-1) times is
divisible by P, where P is a prime number >5.
*Examples:* 222222 is divisible by 7
444444….. 18 times is divisible by 19

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**Base System Concepts
**

**Decimal Binary Hex
**

**0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
8 1000 8
9 1001 9
**

**10 1010 A
11 1011 B
12 1100 C
13 1101 D
14 1110 E
15 1111 F
**

Converting from base ‘n’ to decimal

(pqrst)n = pn4 + qn3 + rn2 + sn + t

Converting from decimal to base ‘n’

# The example given below is converting from 156 to binary. For this we need to keep dividing by 2 till we get the quotient as 0.

2)156 0 2)78 0 2)39 1 2)19 1 2)9 1 2)4 0 2)2 0 2)1 1 0

Starting with the bottom remainder, we read the sequence of remainders upwards to the top. By that, we get 15610 = 100111002

**Tip: ** (pqrst)n x n2 = (pqrst00)n

(pqrst)n x n3 = (pqrst000)n

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**Averages
**

Simple Average = Sum of elements Number of elements

Weighted Average =

Arithmetic Mean = (a1 + a2 + a3 ….an) / n

Geometric Mean =

Harmonic Mean =

For two numbers a and b

AM = (a + b)/2 GM = √����. ����

HM = 2�������� ����+����

** Median** of a finite list of numbers can be found by
arranging all the observations from lowest value to
highest value and picking the middle one.

** Mode** is the value that occurs most often

**Tip: ** AM ≥ GM ≥ HM is always true. They will be
equal if all elements are equal to each other. If I have
just two values then GM2 = AM x HM
**
Tip: ** The sum of deviation (D) of each element with
respect to the average is 0
���� = �����1 − ����������������� + �����2 − ����������������� +

�����3 − �����������������… + �����1 − ����������������� = 0

**Tip: ���������������� **= �������������������������������� ������������ +
������������������������������������

��������.�������� ��������������������������������

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**Percentages
**

Fractions and their percentage equivalents:

**Fraction %age Fraction %age
**

**1/2 50% 1/9 11.11%
**

**1/3 33.33% 1/10 10%
**

**1/4 25% 1/11 9.09%
**

**1/5 20% 1/12 8.33%
**

**1/6 16.66% 1/13 7.69%
**

**1/7 14.28% 1/14 7.14%
**

**1/8 12.5% 1/15 6.66%
**

**Tip: ** r% change can be nullified by
����������������
������������+����

% change in

another direction. Eg: An increase of 25% in prices can be nullified by a reduction of [100x25/(100+25)] = 20% reduction in consumption.

**Tip: **If a number ‘x’ is successively changed by a%,
b%, c%...

Final value = ���� �1 + ���� 100 � �1 + ����

100 � �1 + ����

100 �…

**Tip: ** The net change after two successive changes of

a% and b% is ����� + ���� + ��������100�%

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**Interest
**

Amount = Principal + Interest

Simple Interest = PNR/100

Compound Interest = P(1+ ���� 100

)n – P

Population formula P’ = P(1 ± ���� 100

)n

Depreciation formula = Initial Value x (1 – ���� 100

)n

**Growth and Growth Rates
**

Absolute Growth = Final Value – Initial Value

Growth rate for one year period = Final value – Initial Value

Initial Value x 100

SAGR or AAGR = Final value – Initial Value

No. of years x 100

CAGR= (Final value – Initial Value Initial Value

) 1

������.���� ���������� – 1

**Tip: ** SI and CI are same for a certain sum of money
(P) at a certain rate (r) per annum for the first year.
The difference after a period of two years is given by

∆ = �������� 2

1002

**Tip: **If the time period is more than a year, CAGR <
AAGR. This can be used for approximating the value
of CAGR instead of calculating it.

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**Profit and Loss
**

**%Profit / Loss** =
Selling Price – Cost Price

Initial Value ���� 100

In case false weights are used while selling,

**% Profit** = �Claimed Weigth−Actual Weight Actual Weight − 1� ���� 100

**Discount %** =
Marked Price –Selling Price

Marked Price x 100

**Mixtures and Alligation
Successive Replacement** – Where *a *is the original
quantity, *b *is the quantity that is replaced and *n *is the
number of times the replacement process is carried out,
then

Quantity of original entity after n operation Quantity of mixture

= � a− b

a � n

**Alligation** – The ratio of the weights of the two items
mixed will be inversely proportional to the deviation of
attributes of these two items from the average attribute
of the resultant mixture

Quantity of first item Quantity of second item

= ��2− �� ��− ��1

**Tip: ** Effective Discount after successive discount of

a% and b% is (a + b – �������� 100

). Effective Discount when

you buy x goods and get y goods free is y x+y

x 100.

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**Ratio and Proportion
**

**Compounded Ratio** of two ratios a/b and c/d is ac/bd,
**Duplicate ratio** of a : b is a2 : b2
**Triplicate ratio** of a : b is a3 : b3
Sub-duplicate ratio of a : b is a : b
Sub-triplicate ratio of a : b is ³ a : ³ b
**Reciprocal ratio** of a : b is b : a

**Componendo and Dividendo
**

If ���� ����

= ���� ����

& ���� ≠ ���� ��ℎ���� ����+���� ����−����

= ����+���� ����−����

Four (non-zero) quantities of the same kind a,b,c,d are said to be in proportion if a/b = c/d.

The non-zero quantities of the same kind a, b, c, d.. are said to be in continued proportion if a/b = b/c = c/d.

**Proportion
**a, b, c, d are said to be in proportion if ����

���� = ���� ����

a, b, c, d are said to be in continued proportion if ���� ����

= ���� ����

= ���� ����

Given two variables *x* and *y*, *y is (directly) proportional
to x* (

*x and y*or

**vary directly**,*x and y are in*) if there is a non-zero constant

**direct variation***k*such that y = kx. It is denoted by

Two variables are **inversely proportional** (or **varying
inversely**, or in **inverse variation**, or in **inverse
proportion** or **reciprocal proportion**) if there exists a
non-zero constant *k* such that y = k/x.

**Tip: ** If a/b = c/d = e/f = k

����+����+���� b+d+f

= k

��������+��������+�������� pb+qd+rf

= k

������������+������������+������������

����b���� +����d����+����f���� = kn

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**Time Speed and Distance
**

Speed = Distance / Time

1 kmph = 5/18 m/sec; 1 m/sec = 18/5 kmph

SpeedAvg = ������������������ �������������������������������� �������������������������� ������������������ �������������� ����������������

= ����1+ ����2+ ����3….�������� ����1+ ����2+ ����3….��������

If the distance covered is constant then the average speed is Harmonic Mean of the values (s1,s2,s3….sn)

SpeedAvg = ����

1/����1+ 1/����2+ 1/����3….1/��������

SpeedAvg = 2����1����2 ����1+ ����2

(for two speeds)

If the time taken is constant then the average speed is Arithmetic Mean of the values (s1,s2,s3….sn)

SpeedAvg = ����1+ ����2+ ����3….��������

����

SpeedAvg = ����1+ ����2 2

(for two speeds)

**For Trains,** time taken =
������������������ ��������������������ℎ �������� �������� ����������������������������

�������������������������������� ������������������

**For Boats**,

SpeedUpstream = SpeedBoat – SpeedRiver

SpeedDownstream = SpeedBoat + SpeedRiver

SpeedBoat = (SpeedDownstream + SpeedUpstream) / 2

SpeedRiver = (SpeedDownstream – SpeedUpstream) / 2

**Tip: ** Given that the distance between two points is
constant, then

If the speeds are in *Arithmetic Progression*,
then the times taken are in *Harmonic
Progression*

If the speeds are in *Harmonic Progression*, then
the times taken are in *Arithmetic Progression*

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**For Escalators**,The difference between escalator
problems and boat problems is that escalator can go
either up or down.

**Races & Clocks
**

**Linear Races
**

Winner’s distance = Length of race

Loser’s distance = Winner’s distance – (beat distance + start distance)

Winner’s time = Loser’s time – (beat time + start time)

Deadlock / dead heat occurs when beat time = 0 or beat distance = 0

**Circular Races
**

Two people are running on a circular track of length L with speeds a and b in the same direction

Time for 1st meeting = �� ����−����

Time for 1st meeting at the starting point = LCM ������� ,

�� �����

Two people are running on a circular track of length L with speeds a and b in the opposite direction

Time for 1st meeting = �� ����+����

Time for 1st meeting at the starting point =

LCM ������� , �� �����

Three people are running on a circular track of length L with speeds a, b and c in the same direction Time for 1st meeting = LCM � ������−���� ,

�� ����−���

Time for 1st meeting at the starting point = LCM ������� ,

�� ���� , �� ���

**Clocks **To solve questions on clocks, consider a circular
track of length 360°. The minute hand moves at a speed
of 6° per min and the hour hand moves at a speed of ½°
per minute.

**Tip: ** Hands of a clock coincide (or make 180°) 11 times in every
12 hours. Any other angle is made 22 times in every 12 hours.

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**Time and Work
**

If a person can do a certain task in t hours, then in 1 hour he would do 1/t portion of the task.

A does a particular job in ‘a’ hours and B does the same

job in ‘b’ hours, together they will take �������� ����+����

hours

A does a particular job in ‘a’ hours more than A and B combined whereas B does the same job in ‘b’ hours more than A and B combined, then together they will take √�������� hours to finish the job.

**Tip: ** If A does a particular job in ‘a’ hours, B does the
same job in ‘b’ hours and ABC together do the job in
‘t’ hours, then

C alone can do it in ������������ ��������−��������−��������

hours

A and C together can do it in �������� ����−����

hours

B and C together can do it in �������� ����−����

hours

**Tip: **If the objective is to fill the tank, then the *Inlet
pipes* do **positive work** whereas the *Outlet pipes* do
**negative work**. If the objective is to empty the tank,
then the *Outlet pipes* do **positive work **whereas the
*Inlet Pipes* do **negative work**.

**Tip: **A does a particular job in ‘a’ hours, B does the
same job in ‘b’ hours and C does the same job in ‘c’

hours, then together they will take ������������ ��������+��������+��������

hours.

**Tip: ** If A does a particular job in ‘a’ hours and A&B
together do the job in ‘t’ hours, the B alone will take
��������
����−����

hours.