General Mathematics

Question:Divisibility by 11: A number is divisible by 11,if the difference of the sum of its digits  at odd places and the sum of its digits at at even places,is either o or a number divisible by 11.

10.Show that 4832718 is divisible 11. 
AnswerSol.( Sum of digits at odd places)-(Sum of digits at even places)
     =[(8+7+3+4)-(1+2+8)=11,which is divisible by 11.
      :.4832718 is divisible by 11.
  Co-Prime:Two numbers are said to be co-prime,if their H. C. F. is 1.
e.g.(2,3),(4,5),(7,9),(8,11) etc.are primes.

An Important Note: A number is divisible by ab only when that number is divisible by each one of a and b,where a and b are co-prime.

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Question:11. If a number is divisible by both 4 and 6,its always divisible by 24?
why ?Give an example. 
AnswerSol.No,since 6 and 4 are not co-prime.
    36 is divisible by 6 as well as 4 but it is not divisible by 24.

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Question:12.Without actual division show that 52563744 is divisible by 24. 
AnswerSol.24=3x8, where 3 and 8 are co-prime.
The sum of the digits in given number is 36,which is divisible by 3
So the given number is divisible by 3.
The number formed by the last 3-digits of the given number  is 744,which is divisible by 8.
 So,the given number is divisible by 8.
Thus,the given number is divisible by both 3 and 8,where 3 and 8 are co-prime.
 So,it is divisible by 3x8,i.e.24.


FORMULAE:(i)`(a^2+b^2)`=`a^2+b^2+2ab`
                 
                (ii) `(a-b)^2`=`a^2+b^2-2ab`
                 
                (iii) `(a+b)^2-(a-b)^2`=4ab
                  
                (iv) `(a+b)^2+(a-b)^2`=`2(a^2+b^2)`
                  
                (v) `(a^2-b^2)`= (a+b)(a-b)
                  
                (vi) `(a^3+b^3)`= (a+b)`(a^2-ab+b^2)`
                  
                (vii) `(a^3-b^3)` = `(a-b) (a^2+ab+b^2)`
                  
                (viii) a.(b+c)=ab+ac θ a.(b-c)=ab-ac (Distributive Laws)

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Question:13. Multiply 5793405 by 99999 by short cut methed . 
AnswerSol. `(i) 5793405 xx 99999 = 5793405 xx (100000 -1)`
                                 
                       = `(579340500000 - 5793405)`
                                  
                        =579334706595.

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Question:14.Evalute : 
        (i) `986 xx 137 +986 xx 863`     

        (ii) `983 xx 207 - 983 xx 107` 
AnswerSol. `(i)986 xx 137 + 986 xx 863 = 986(137+863)`
                                     
                    `=986 xx 1000 = 986000.`
       
      `(ii) (983 xx 207 - 983 xx 107 = 983 xx (207 - 107)`
                                       
                       `=983 xx 100 = 98300.`

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Question:Divisibility by 11: A number is divisible by 11,if the difference of the sum of its digits at odd places and the sum of its digits at at even places,is either o or a number divisible by 11. 10.Show that 4832718 is divisible 11.

Question:11. If a number is divisible by both 4 and 6,its always divisible by 24? why ?Give an example.

Question:12.Without actual division show that 52563744 is divisible by 24.

Question:13. Multiply 5793405 by 99999 by short cut methed .

Question:14.Evalute : (i) `986 xx 137 +986 xx 863` (ii) `983 xx 207 - 983 xx 107`