Arithmetic Quick Refresh, Exercises of Mathematics

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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 (pq)

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.

HCF and LCM

For two numbers, HCF x LCM = product of the two.

HCF of Fractions = (^) 𝐿𝐶𝑀𝐻𝐶𝐹 𝐷𝐷𝑜𝑜^ 𝐷𝐷𝑜𝑜 𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑟𝑟^ 𝑁𝑁𝑎𝑎𝑎𝑎𝑎𝑎𝑟𝑟𝑎𝑎𝐷𝐷𝐷𝐷𝑟𝑟

LCM of Fractions = (^) 𝐻𝐶𝐹𝐿𝐶𝑀 𝐷𝐷𝑜𝑜^ 𝐷𝐷𝑜𝑜 𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑎𝑎𝐷𝐷𝐷𝐷𝑟𝑟^ 𝑁𝑁𝑎𝑎𝑎𝑎𝑎𝑎𝑟𝑟𝑎𝑎𝐷𝐷𝐷𝐷𝑟𝑟

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 = x a^ ybzc^ 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 =

xa+1^ – 1 𝑥−1 𝑋^

y b+1^ – 1 𝑦−1 𝑋^

z c+1^ – 1 𝑧− 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 ∅(𝑁) = 𝑁 � 1 −

1 𝑥 � �^1 −^

1 𝑦 � �^1 −^

1 𝑧 �^ {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

a 3 ± b 3 = (a ± b)(a^2 ∓ ab + b^2 ). Hence, a^3 ± b^3 is divisible by (a ± b) and (a^2 ± ab + b^2 ).

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

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

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

a 3 + b 3 + c 3 - 3abc = (a + b + c)(a 2 + b 2 + c^2 - ab - ac - bc) Hence, a^3 + b^3 + c 3 = 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.

Remainder Related Theorems

Euler’s Theorem :

Number of numbers which are less than N = 𝑎𝑎 𝑝𝑝^ ∗ 𝑏𝑏 𝑞𝑞^ ∗ 𝑐 𝑟𝑟 and co-prime to it are

 ∅(𝑁) = 𝑁 � 1 −

1 𝑎𝑎 � �^1 −^

1 𝑏𝑏 � �^1 −^

1 𝑐𝑐 �

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

 𝑹𝒆𝒎 � 𝑴^

∅(𝒏) 𝑵 �^ =^ 𝟏𝟏

Example: 𝑅𝑒𝑚 �

750 90 �^ =?

 ∅(90) = 90 � 1 −

1 2 � �^1 −^

1 3 � �^1 −^

1 5 �  ∅(90) = 90 ∗ 12 ∗ 23 ∗ 45 = 24

 𝑅𝑒𝑚 � 7

24 90 �^ = 1 =^ 𝑅𝑒𝑚 �^

748 90 �

 𝑅𝑒𝑚 �

750 90 �^ =^ 𝑅𝑒𝑚 �^

72 90 � ∗ 𝑅𝑒𝑚 �^

748 90 �^ = 49^ ∗^ 1 = 49

Fermat’s Theorem: If N is a prime number and M and N are co-primes  𝑹𝒆𝒎 � 𝑴^

𝑵 𝑵 �^ =^ 𝑴

 𝑹𝒆𝒎 � 𝑴^

𝑵−𝟏𝟏 𝑵 �^ =^ 𝟏𝟏

Example: 𝑅𝑒𝑚 �

631 31 �^ = 6 &^ 𝑅𝑒𝑚^ �^

630 31 �^ = 1

Wilson’s Theorem If N is a prime number  𝑹𝒆𝒎 �

(𝑵−𝟏𝟏)! 𝑵 �^ =^ 𝑵 − 𝟏𝟏

 𝑹𝒆𝒎 �(𝑵−𝟐 𝑵 )!� = 𝟏𝟏 Example: 𝑅𝑒𝑚 � 3031! � = 30 & 𝑅𝑒𝑚 � 2931! � = 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 = pn^4 + qn^3 + rn^2 + 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 156 10 = 10011100 (^2)

Tip: (pqrst) (^) n x n^2 = (pqrst00)n

(pqrst)n x n^3 = (pqrst000)n

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

𝟏𝟏𝟏𝟏𝟏𝟏𝟏𝟏 𝟏𝟏𝟏𝟏𝟏𝟏+𝟏𝟏

% 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 = 𝑥𝑥 �1 + 100 𝑎𝑎 � �1 + 100 𝑏𝑏 � �1 + 100 𝑐𝑐 � …

Tip: The net change after two successive changes of

a% and b% is �𝑎𝑎 + 𝑏𝑏 + 100 𝑎𝑎𝑏𝑏� %

Interest

Amount = Principal + Interest

Simple Interest = PNR/

Compound Interest = P(1+ 100 𝑟𝑟 )

n

• P

Population formula P’ = P(1 ± 100 𝑟𝑟 )n

Depreciation formula = Initial Value x (1 –

𝑟𝑟 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 𝑛𝑛𝑜.𝑜𝑓 𝑦𝑒𝑎𝑟𝑠 (^) – 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

𝑃𝑃𝑅𝑅^2 100 2

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.

Ratio and Proportion

Compounded Ratio of two ratios a/b and c/d is ac/bd, Duplicate ratio of a : b is a^2 : b^2 Triplicate ratio of a : b is a^3 : b^3 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

𝑎𝑎 𝑏𝑏 =^

𝑐𝑐 𝑎𝑎 &^ 𝑎𝑎 ≠ 𝑏𝑏^ 𝑡ℎ𝑒𝑛^

𝑎𝑎+𝑏𝑏 𝑎𝑎−𝑏𝑏 =^

𝑐𝑐+𝑎𝑎 𝑐𝑐−𝑎𝑎 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

𝑎𝑎 𝑏𝑏 =^

𝑐𝑐 𝑎𝑎 a, b, c, d are said to be in continued proportion if 𝑎𝑎 𝑏𝑏 =^

𝑏𝑏 𝑐𝑐 =^

𝑐𝑐 𝑎𝑎

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

𝑝𝑝𝑎𝑎 𝑛𝑛^ +𝑞𝑞𝑐𝑐 𝑛𝑛^ +𝑟𝑟𝑎𝑎𝑛𝑛

𝑝𝑝b 𝑛𝑛^ +𝑞𝑞d 𝑛𝑛^ +𝑟𝑟f 𝑛𝑛^

= kn

Time Speed and Distance

Speed = Distance / Time

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

SpeedAvg =

𝑇𝐷𝐷𝐷𝐷𝑎𝑎𝑒𝑒 𝐷𝐷𝐷𝐷𝑎𝑎𝐷𝐷𝑎𝑎𝐷𝐷𝑐𝑐𝑎𝑎 𝐶𝐷𝐷𝑎𝑎𝑎𝑎𝑟𝑟𝑎𝑎𝑎𝑎 𝑇𝐷𝐷𝐷𝐷𝑎𝑎𝑒𝑒 𝑇𝐷𝐷𝑎𝑎𝑎𝑎 𝑇𝑎𝑎𝑘𝑎𝑎𝐷𝐷 =^

𝑎𝑎 1 + 𝑎𝑎 2 + 𝑎𝑎 3 ….𝑎𝑎𝑛𝑛 𝐷𝐷 1 + 𝐷𝐷 2 + 𝐷𝐷 3 ….𝐷𝐷𝑛𝑛

If the distance covered is constant then the average speed is Harmonic Mean of the values (s 1 ,s 2 ,s 3 ….s (^) n)

 SpeedAvg =

𝐷𝐷 1 /𝑎𝑎 1 + 1 /𝑎𝑎 2 + 1 /𝑎𝑎 3 …. 1 /𝑎𝑎𝑛𝑛

 SpeedAvg =

2𝑎𝑎 1 𝑎𝑎 2 𝑎𝑎 1 + 𝑎𝑎 2

(for two speeds)

If the time taken is constant then the average speed is Arithmetic Mean of the values (s 1 ,s 2 ,s 3 ….s (^) n)

 SpeedAvg =

𝑎𝑎 1 + 𝑎𝑎 2 + 𝑎𝑎 3 ….𝑎𝑎𝑛𝑛 𝐷𝐷  SpeedAvg =

𝑎𝑎 1 + 𝑎𝑎 2 2

(for two speeds)

For Trains, time taken =

𝑇𝐷𝐷𝐷𝐷𝑎𝑎𝑒𝑒 𝑒𝑒𝑎𝑎𝐷𝐷𝑎𝑎𝐷𝐷ℎ 𝐷𝐷𝐷𝐷 𝑏𝑏𝑎𝑎 𝑐𝑐𝐷𝐷𝑎𝑎𝑎𝑎𝑟𝑟𝑎𝑎𝑎𝑎 𝑅𝑅𝑎𝑎𝑒𝑒𝑎𝑎𝐷𝐷𝐷𝐷𝑎𝑎𝑎𝑎 𝑆𝑝𝑝𝑎𝑎𝑎𝑎𝑎𝑎 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

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

𝑎𝑎𝑏𝑏 𝑎𝑎+𝑏𝑏 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 √𝑎𝑎𝑏𝑏 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 (^) 𝑎𝑎𝑏𝑏−𝑎𝑎𝑎𝑎𝑏𝑏𝐷𝐷𝐷𝐷−𝑏𝑏𝐷𝐷 hours

 A and C together can do it in (^) 𝑏𝑏−𝑏𝑏𝐷𝐷𝐷𝐷 hours

 B and C together can do it in

𝑎𝑎𝐷𝐷 𝑎𝑎−𝐷𝐷 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

𝑎𝑎𝑏𝑏𝑐𝑐 𝑎𝑎𝑏𝑏+𝑏𝑏𝑐𝑐+𝑐𝑐𝑎𝑎 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 𝑎𝑎𝐷𝐷 𝑎𝑎−𝐷𝐷 hours.