
UN celebrating 2012 IYC (International Year of Cooperatives)

UN Celebrating 2011 International Year of Forest and Youth

Mental Maths

Numbers

Prime Numbers: A number greater than 1 is called a prime number, if it has exactly two factors, namely 1 and the number itself.

Prime Numbers greater than 100 : let p be a given number greater than 100 To find out whether it is prime or not, we use the following method:
Find a whole number nearly greater than the square root of p. Let k > p. Test whether p is divisible by any prime number less than k. If yes, then p is not a prime number. Otherwise p is prime number
Ex. We have to find whether 191 is a prime number or not. Now, 14 > 191 Prime Numbers less than 14 are 2,3,5,7,11,13. 191 is not divisible by any of them. So 191 is a prime number.

Composite numbers: Numbers greater tha 1 which are not prime, are known as composite is number

Note: 1 is neither prime nor composite

2 is the only even number which is prime

There are 25 prime numbers b/w 1 and 100

Co primes: Two number a and b are said to be coprimes, if their HCF is 1 eg. (2,3), (4,5)

A number is divisible by 2 : if its unit’s digit is any of 0,2,4,6,8

Divisibility by 3: if the sum of its digits is divisible by 3

Divisibility by 4: If the number formed by last two digit is divisible by 4

Divisibility by 5: If the unit digit is either 0 or 5

Divisibility by 6: If it is divisible by 2 and 3

Divisibility by 8: If the number formed by the last three digits of the given number is divisible by 8

Divisibility by 9: If the sum of its digits are divisible by 9

Divisibility by 10: if the number ends with 0

Divisibility by 11: (sum of the digits at odd place)(even place) is either 0 or divisible by 11

Divisibility by 12: If divisible by 3 and 4

Divisibility by 14: If divisible by 2 and 7

Divisibility by 15 : if divisible by 3 and 5

Divisibility by 16: If the number formed by last for digits is divisible by 16

Divisibility by 24: If divisible by both 3 and 8

Divisibility by 40: if it is divisible by both 5 and 8

Divisibility by 80: if it is divisible by both 5 and 16

If a number is divisible by p as well q, where p and q are coprimes, then the given number is divisible by pq
If p and q are not coprimes then the given number need not be divisible by pq even when it is divisible by both p and q

Multiplication by SHORT CUT METHODS

Multiplication by Distributive Law:
a x (b+c) = axb + axc
a x (bc) = axb  axc

Multiplication of a Number by 5^{n} : Put n zeros to the right of the multiplicand and divide the number so formed by 2^{n}
Example : 975436 x 625 = 975436 x 5^{4} = 9754360000/16

(a+b)^{2} = a^{2} + b^{2} + 2ab

(ab)^{2} = a^{2} + b^{2}  2ab

(a+b)^{2 }–(ab)^{2} = 4ab

(a+b)^{ 2} + (ab)^{ 2} = 2(a^{2 }+b^{2})

(a^{2}_{ } b^{2}) = (a_{+}b) (ab)

(a+b+c)^{ 2} = a^{2} + b^{2} + c^{2} + 2 (ab+ bc+ca)

(a^{3} + b^{3}) = (a+b)(a^{2}ab+b^{2})

(a^{3} –b^{3}) = (ab) (a^{2}+ab+b^{2})

(a^{3}+b^{3}+c^{3} 3abc) = (a+b+c)(a^{2}+b^{2}+c^{2}abbcca)

If a + b + c = 0 , then a^{3}+b^{3}+c^{3} = 3abc

1 +2+3+…..+n = n(n+1)/2

1^{2}+2^{2}+3^{2}+….+n^{2} = n(n+1)(2n+1)/6

1^{3}+2^{3 }+3^{3}….+n^{3 }= n^{2}(n+1)^{2}/4

HCF AND LCM OF NUMBERS

Factors and Multiples: If a number a divides another number b exactly, we say that a is a factor of b. In this case, b is called a multiple of a

HCF or Greatest Common Measure or Greatest Common Divisor: is the greatest number that divides each of them exactly.

Factorization Method: Express each one of the given numbers as the product of Prime Factors. The product of least powers of common prime factors gives HCF

Division Method: Suppose we have to find the HCF of two given numbers. Divide the larger number by the smaller one. Now, divide the divisor by the remainder. Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder. The last divisor number is required HCF.

Finding the HCF of more than two numbers: Suppose we have to find the HCF of three numbers. Then, HCF of (any two ) and the third numbers gives the HCF of three numbers.

Least Common Multiple: The least number which is exactly divisible by each one of the given numbers is called their LCM

Factorization Method: Resolve each one of the given numbers into a product of prime factors. Then, LCM is the product of highest powers of all the factors.

Common Division Method: Arrange the given numbers in a row in any order. Divide by a number which divides exactly at least two of the given numbers and carry forward the numbers which are not divisible. Repeat the above process till no two of the numbers are divisible by the same number except 1 The product of the divisors and the undivided numbers is the required LCM of the given numbers.

Products of two numbers = Product of their HCF and LCM

HCF and LCM of Fractions:

HCF = HCF of Numerators/ LCM of denominators

LCM = LCM of Numerators/HCF of denominators

HCF and LCM of Decimal Fractions: In a given numbers, make the same number of decimal places by annexing zeros in some numbers, if necessary. Considering these numbers without decimal point, find HCF or LCM as the case may be. Now, in the result, mark off as many decimal places as are there in each of the given numbers

Comparison of fraction: Find the LCM of the denominators of the given fractions, Convert each of the fractions into an equivalent fractions with LCM as the denominator by multiplying both the numerator and denominator by the same number. The resultant fraction with the greatest numerator is the greatest.
Suppose a man covers a certain distance at x km/hr & an equal distance at y km/hr
Average speed = 2xy/(x+y)
Average of First n natural number = n(n+1)/2n = n+1/2
Mean of square of 1^{2} 2^{2}, 3^{2} ……n^{2} = n(n+1)(2n+1)/6
ABOUT INDIA

Longest National Highway : No. 07

National Highway 1 and 2 : Grand Trunk Road

Highest Roadway of World : LehManali

State with largest no. of roads : Maharashtra

Largest Pacca Road : Maharashtra

Largest Kaccha Road : Odisha

Max Road Denisty and Minimum : Goa and J & K

First Rail in India : April 1853 (Bombay to Thane)

First Rail in World : 1825 (Liverpool to Manchester)

Railway Board in India : 1905

Nationalization of Indian Railway : 1950

Underground Railway : 24 October 1984 Kolkata

First Electric Train : Deccan Queen PUne

Air services in India : 1911

Air mail services in India : Allahabad to Naini

Natural Harbor in India : Vishakhapatnam

Kandal is a tidal port

Population Census : 7 Schedule Article 69

Lord Rippon started the census in year 1881

Population Density in India : 325/ sq. km

Highest population density state : West Bengal

Lowest density state : Arunanchal Pradesh

Highest population : Uttar Pradesh

Lowest population : Sikkim

Highest literacy : Keral

Lowest literacy : Bihar

Highest urban population : Goa

Lowest urban population : Himanchal Pradesh

First national park : Corbet national park (Helly) UP

Highest national park state : MP (11) Tiger state

India’s largest national park : Himis

Siberian Birds : Kevla Dev Ghana Bird Sanctuary Rajasthan

Abode of God : Prayag

Land of Five Rivers : Punjab

City of Seven Mountains : Mumbai

City of Handicrafts : Panipat

Venice of East : Kochi

Garden of Spices : Kerala

Heaven of Fruit Garden : Sikkim

Detrite of India : Pithampura

Paris of East : Jaipur

City of Festivals : Madurai

Queen of Mountains : Missouri

Queen of Arab Ocean : Kochi

Scotland of East : Meghalaya

City of Mountains : Dungarpur

Country of Malaya : Karnatak

Most polluted river : Sabarmati

Ganges of Dakshin Bharat : Kaveri

Black River : Sharda

Egg basket of Asia : Andhra Pradesh

Heart of Rajasthan : Ajmer

Perfumes City : Kanuaj

Sister of Kashi : Gazipur

City of Lettuce : Dehradun

Super Developed City : Chennai

Old Ganges : Godavari

Sorrow Of West Bengal : Damodar

Sun City : Jodhpur

Pride of Rajasthan : Chittodgadh

City of Coal : Dhanbad

Battle of Plassey : 1757

Battle of Buxar : 1764

Regulating Act : 1773

Pitts India Act : 1784

Bharat Shashan Adhiniyam : Morley Mintto Reform 1909

Montague Chelmsford : 1919

Largest unit of Distance Measurement Parsec (3.26 Light Year)

CGS Unit : Dyne MKS Newton

Newton Law of Motion : 1687 Principia

Momentum : Mass x Acceleration

Second Type of Liver : F W E

Third Type of Liver : F E W

Kinetic Energy = P^{2}/2m ( P = momentum x velocity / m = Mass)

Power = Joule / Second = Work / time = watt

