Numbers, General Mathematics

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:15.Evaluate :
 (i)`(527 xx 527 xx 527 + 183 xx 183 xx 183)/(527 xx 527 - 527 xx 183 xx 183 xx 183)`

 (ii)`(458 xx 458 xx 458 - 239 xx 239 xx 239)/(458 xx 458 + 458 xx 239 + 239 xx 239)`

 (iii)`((614 +168)^2-(614-168)^2/614 xx 168)`

 (iv)`((832+278)^2+(832-278)^2/832 xx 832 + 278 xx 278)` 
AnswerSol.(i) Given Expression =`(527)^3 + (183)^3/(527)^2 - 527 xx 183 + (183)^2`
                                  
                                   =`a^3 + b^3/a^2 - ab + b^2`
                                  
                                   = (a+b) = (527 + 183) =710

      (ii) Given Expression =`(458)^3 - (239)^3/(458)^2+458 xx 239 + (239)^2`
                                    
                                    =`(a^3-b^3)/(a^2+ab+b^2)`
                                    
                                   = (a-b) = (458-239) = 219.
    
      (iii) Given Expression =`(a+b)^2-(a-b)^2)/ab`
    
      (iv) Given Expression =`((a+b)^2+(a-b)^2)/(a^2+b^2)`
                                    
                                     =`(2(a^2+b^2)/(a^2+b^2))`=2

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Question:16.Simplify :

   (i) `1605 xx 160 =?`

`(ii) 1398 xx 1398 =?` 
AnswerSol. (i)`1605 xx 1605 = (1605)^2`
                            =`(1600+5)^2`
                             =`(1600)^2+5^2+2x1600x5`
                              =2560000+25+16000
                              =2576025

     (ii)1398x1398=`(1398)^2`
                         =`(1400-2)^2`
                         =`(1400)^2+2^2-2x1400x2`
                         =1960000+4-5600
                          =1954404.

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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`

Question:15.Evaluate : (i)`(527 xx 527 xx 527 + 183 xx 183 xx 183)/(527 xx 527 - 527 xx 183 xx 183 xx 183)` (ii)`(458 xx 458 xx 458 - 239 xx 239 xx 239)/(458 xx 458 + 458 xx 239 + 239 xx 239)` (iii)`((614 +168)^2-(614-168)^2/614 xx 168)` (iv)`((832+278)^2+(832-278)^2/832 xx 832 + 278 xx 278)`

Question:16.Simplify : (i) `1605 xx 160 =?` `(ii) 1398 xx 1398 =?`