Basic Number Theory

• Square of every even Number = An even Number

• Sum of first ‘n’ natural numbers = {n(n+1)}/2

• Sum of first ‘n’ odd numbers = n²

• Sum of first ‘n’ even numbers = n(n+1)

• Sum of square of first ‘n’ natural numbers = {n(n+1)(2n+1)}/6

• Sum of cubes of first ‘n’ natural numbers = {n(n+1)/2}²

• There are 15 prime numbers between 1 & 50 and 10 prime numbers between 50 & 100

• If n divides p & q, then n divides their sum and difference also

      Like: 6 divides 12 & 18, then 12+18 = 30 and 18-12 = 6 are also divided by 6

• For any natural number n, (n³ – n) is divisible by 6

• The product of three consecutive natural numbers is always divisible by 6

• (x^m – a^m) is divisible by (x – a) for all values of m

• (x^m – a^m) is divisible by (x + a) for even values of m

• (x^m + a^m) is divisivlke by (x + a) for odd values of m

• Number of prime factors of a^p b^q c^r d^s is p+q+r+s, where a, b, c, d are prime numbers

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How to find Unit Place Digit Phase 2

We have already found the units place digit for some numbers like 0, 1, 5, 6 and 2. So far we have seen that there are some numbers for which we don’t need any explanation to find out their units place digit.

Here we come to find the units place digit of 3, 4, 8, and 7.

Let’s take a look:

If Units place digit is 4 or 8, then we should first divide the power by 4 and then represent in the form of 4⁴ or 8⁴. In that case the value of 4⁴ and 8⁴ will be 6. [Because 4*4*4*4 = 256 or 8*8*8*8 = 4096]

       (124)³⁷²    = (4) ^(372/4)  

                         = (4⁴)⁹³

                         = 4⁴ = 256                                   

       (128)³²⁵     = (8) ^(324/4)

                          = (8⁴)⁸¹

                          = 8⁴ = 4096

So the unit place digit is 6.

Again, if the unit place digit is 3 or 7, then we should first divide the power by 4 and then represent in the form of 3⁴ and 7⁴. In that case the value of 3⁴ and 7⁴ will be 1.                                                                  

        (2467)¹⁵³   = (7) ^(153/4)                    

                          = (7⁴)³⁸ * 7¹                                 

                           = 7⁴ * 7                                        

                          = 1 * 7 = 7                                  

                      [7⁴ = 7*7*7*7 = 2401]                               

       (293)¹⁴¹     = (3) ^(141/4)

                          = (3⁴)³⁵ * 3¹

                          = 3⁴ * 3

                          = 1 * 3 = 3

                      [3⁴ = 3*3*3*3 = 81]

 So the unit place digits are 7 and 3 respectively.

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How to find Unit place Digit

Finding Unit place digit is very easy when it comes to multiplication of nay two or more numbers. Multiply the last digits of the numbers and you get the result. But what if a number is given with power of two or three or more digits to it. It’s not as hard as you imagine.

Let’s take a look:

If the unit place digit is 0 or 1 or 5 or 6 then the resultant Unit place digit remains same.

        (479)¹¹⁵¹   =  Its Unit digit is 6.

        (155)¹²⁰    =   Its Unit digit is 5.

        (191)¹⁹     =   Its Unit digit is 1.

        (900)⁵¹     =   Its unit digit is 0.

If Unit place digit is 2, then the power of the number is first divided by 4 and there after represented in the form of 2⁴

(572)⁴⁴²    =   (2) ^(442/4) [taking unit place digit or the last digit of 572] 

                  =   (2⁴)¹¹⁰ * 2²

                  =   2⁴ * 2²

                  =   16 * 8

There taking the multiplication the Unit place digit is 8.  (16*8=48)

To find Unit Place Digits of other numbers visit: Unit Place Digit Phase 2

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Daily GK update 14th May 2018

1. Petrol price hiked after 19 days:

Ending the 19 days freeze on the hike in petrol and diesel prices, the oil marketing companies on Monday May 14, 2018 increased the petrol price by 17 paisa a liter while the diesel price was hiked by 21 paisa.

2. Fortis board went against Financial Advisers on Munjal-Burman bid:

The much wooed Fortis Healthcare may have selected a suitor at last. The recent selection of the winning bid did not get the nod of FHL’s independent directors and as a result the fissure in the boardroom has widened further.

3. WPI inflation shoots to 3.18% in April on costlier fuel:

Inflation based on whole sale price index spiked to 3.18% per cent in April on increasing prices of petrol and diesel as well as fruits and vegetables. The whole sale price index stood at 2.47 per cent in March and 3.85 per cent in April last year.

4. Walmart may launch IPO for Flipkart:

Walmart Inc said on Saturday in a filing with a U.S. regulator that it may take India’s Flipkart public in as early as four years, detailing for the first time a potential listing time line for Walmart’s largest ever acquisition.

5. Finance Ministry mulling innovative ways to deal with banks’ NPA provision:

The FM is examining a proposal to find innovative ways for dealing with burden of NPA provisions by issuing provision Shore-up certificates to banks.

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Square of 666 Or Square of 666

The traditional method of finding Squares has now become extinct. There are lot more easier way to find out the Squares. We are going to find out the Square of 666 or 6666 with a lot more easiest way.

Let’s take a look:

(666)² = 443556 

(6666)² = 44435556

Note: To find out this square we have used and what you always have to keep in mind is - ‘FOTFIS’. Where,

‘FO’ - Four or 4 ; ‘T’ – Three or 3 ; ‘FI’ – Five or 5 ; ‘S’ – Six or 6.

In the answer the number of digit 4 and the number of digit 5 will be same and the number of each of these digits 4 & 5 will be one less than the number of digit 6 in the question. Again the number of digits 3 & 6 will be one and they will be placed just as shown in the example.

For more visit : AcademiaBlogs

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Multiplication of Consecutive Even/Odd Numbers with gap of 2 or 4 or 6

Here are some new tricks on Multiplication of two consecutive even or odd numbers where the difference in between theme is 2 or 4 or 6 etc.

Let’s take a look:

   Multiplication with gap of 2: 

12 * 14                29 * 31

             (13)²-1                (30)²-1

           

            =169-1                =900-1

            =168                   =899

   Multiplication with gap of 4:      

        12 * 16                     36 * 40

         (14)²-(14-12)²         (38)²-(38-36)²

        

         =196-4                     =1444-4

         =192                        =1440

    Multiplication with gap of 6: 

         12 * 18                     32 * 38

          (15)²-(15-12)²         (35)²-(35-32)²

          

          =225-9                     =1225-9

          =216                        =1216

For quick multiplication remember squares up to 50.

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Daily GK Update 12th May 2018

    

     1.   World’s Largest Ecommerce deal took place between Walmart and Flipkart:

US based eCommerce giant Walmart has confirmed that it will buy 77% stake of Indian origin eCommerce firm Flipkart for $16 billion.

2.     According to Forbes The most powerful people in the World:

PM Modi placed at 9, Chineese President XI Jingping Tops the list. Chineese President is followed by Russsian President Vladimir Putin and US president Donald Trump.

3.     NASA and Uber tie up to make the flying cars reality:

The National Aeronautics and Space Administration signed a space act agreement with Uber technologies to explore concepts and technologies related to Uber Air Mobility for future air transportation.

4.     September 2019 climate summit to be held in New York:

UN Secretary General Antonio Guterres announced to hold a climate summit in September 2019 in New York to review the commitment of 2015 Paris Agreement on Climate Change.

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Squares of 333, Squares of 3333

Finding the square of 333 or 3333 is as simple as to find the square of 222 or 2222. But in that case you have to multiply a larger number. Instead we have made it simpler than that of finding 222.

Lets’ take a look:

 (333)² = 110889    ,    
 (3333)² = 11108889

[Note: To find out square of 33 or 333 or 3333 keep in mind this – “OZEN’, where

O = One or 1 ;  Z = Zero or 0 ; E = Eight or 8 ; N = Nine or 9.

The number of digit one or 1 and the number digit of eight or 8 will be same and the total number of each of these (1 & 8) digits will be one less than the number of 3s in the given calculation. Again in the solution the number of each digits (0 & 9) will be one just as shown in the example.]

To know more visit:
https://academiablogs.blogspot.in/

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Square of Two Digits, Three digits of same number

Find the squares of same numbers as fast as it can be done. Time is limited, so we need to utilize it.

Lets’ take a look:

(11)² = 121.

The square of 11 is known to everyone. But what if the number of digits of 1 increases like 111 or 1111 or 11111? No problem. Here are examples and solutions on the squares of 111 or squares of 1111 or squares of 11111.

(111)² = 12321     ,     (1111)² = 1234321    ,      (11111)² = 123454321

Here are some other example on the Squares of 22 or Squares of 222 and so on.

(22)² = {2*(11)}² = 4*(11)² = 4*(121) = 484 ;

(222)² = {2*(111)}² = 4*(111)² = 4*(12321) = 49284

For more tricks visit: https://academiablogs.blogspot.in/

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Daily GK Update 11th May 2018

1. Nation Technology Day: 11 May

     National Technology Day is celebrated to memorize the anniversary of “Shakti: The Pokhran Nuclear Test” Which was held in 11^th of May 1998

2. RBI puts Dena Bank under Corrective Action:

The public sector lender DENA BANK on 11^th of May 2018 reported the widening of its net loss to Rs. 1,225.42 crore in the March quarter report. The net loss stood at Rs 575.26 crore in the January-March quarter of 2016-17.

3. China’s Alibaba buys Pakistani E-commerce Daraz:

Chinese E-commerce firm Alibaba has purchase Pakistani E-commerce firm Daraz to continue to expand its overseas presence in South Asian Consumer market.

4. DFS to approach revenue dept over GST:

Department of Financial Services (DFS) has decided to approach the revenue department to clarify the confusion on levy of GST on some free services provided by Banks. Financial Services Secretary Rajiv Kumar said “The issue of imposition of GST on some free services that banks provide to its customers will be taken up with the revenue department and we hope to resolve the issue soon.”

5. Bank Unions on 48-hour strike from May 30:

The association of banking union has threatened to go on 48=hours strike from 30^th of May to 31^st of May over the issue of Indian Bank’s Association’s measly wage hike offer and delay in wage negotiation.

For more visit: https://academiablogs.blogspot.in/

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Types of Numbers

• Natural Numbers:

       Natural numbers are counting numbers. They are denoted by ‘N’.

                 N = {1, 2, 3, …} 

• Whole Numbers:

      All natural numbers and zero form the set of whole numbers. Whole numbers are denoted by ‘W’.

      W = {0, 1, 2, 3, …}

• Integers:

      Whole numbers and negative numbers form the set of integers. They are denoted by ‘I’.

       I = {…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …}

• Even Numbers:

      A counting number which is divisible by 2, is called an even number.

     0, 2, 4, 6, 8, 10, … etc.

• Odd Numbers:

      A counting number which is not divisible by 2, is known as ODD number.

      1, 3, 5, 7, … etc.

• Prime Numbers:

      A counting number is called a prime number when it is exactly divisible by 1 and itself.

      2, 3, 5, 7, 11, 13, … etc.

• Composite Numbers:

      Composite numbers are non-prime natural numbers. They must have at least one factor apart from  and itself.

       4, 6, 8, 9 etc.

• Coprimes:

      Two natural numbers are said to be Coprimes, if their HCF is 1.

      (7, 9), (15, 16)

• Rational Numbers:

      A number that can be expressed as p/q is called a rational number, p and q are integers and p ≠ 0.

      3/5, 7/9, 8/9 etc.

• Irrational Numbers:

      The numbers that cannot be expressed in the form of p/q are called irrational

and q are integers and p ≠ 0.

      √2, √3, √7 etc.

• Real Numbers:

      Real Numbers include rational and irrational numbers both.

      7/9, √2, √5, ∏, 8/9 etc.

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