# Arithmetic Quick Refresh, Exercises for Mathematics

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Math Formulas & Fundas

Shortcuts, Formulas & Tips

Vol. 1: Number System & Arithmetic

present

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.

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.

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)

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 divisibleFor 13: 31 + 2 x 4 = 31 + 8 = 39 Divisible.  For 19: 31 + 2 x 2 = 31 + 4 = 35 Not divisible.

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 >>

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

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

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: ���������������� = �������������������������������� ������������ + ������������������������������������

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

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�%

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.

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.

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

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

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.

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.