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  • 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 co-primes, 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 co-primes, then the given number is divisible by pq

If p and q are not co-primes 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

  1. Multiplication by Distributive Law:

a x (b+c) = axb + axc

a x (b-c) = axb - axc



  1. Multiplication of a Number by 5n : Put n zeros to the right of the multiplicand and divide the number so formed by 2n

Example : 975436 x 625 = 975436 x 54 = 9754360000/16

  • (a+b)2 = a2 + b2 + 2ab

  • (a-b)2 = a2 + b2 - 2ab

  • (a+b)2 –(a-b)2 = 4ab

  • (a+b) 2 + (a-b) 2 = 2(a2 +b2)

  • (a2 - b2) = (a+b) (a-b)

  • (a+b+c) 2 = a2 + b2 + c2 + 2 (ab+ bc+ca)

  • (a3 + b3) = (a+b)(a2-ab+b2)

  • (a3 –b3) = (a-b) (a2+ab+b2)

  • (a3+b3+c3 -3abc) = (a+b+c)(a2+b2+c2-ab-bc-ca)

  • If a + b + c = 0 , then a3+b3+c3 = 3abc

  • 1 +2+3+…..+n = n(n+1)/2

  • 12+22+32+….+n2 = n(n+1)(2n+1)/6

  • 13+23 +33….+n3 = n2(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.

  • Average

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 12 22, 32 ……n2 = 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 : Leh-Manali

  • 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 = P2/2m ( P = momentum x velocity / m = Mass)

  • Power = Joule / Second = Work / time = watt



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