Question:Divisiblity By 9:A number is divisible by 9,if the sum of its digits is divisible by 9.

7.Which of the following numbers is divisible by 9.
 (i)19725462 (ii) 36870521 

Sol.(i) Sum of digits in 19725462 is 36,which is divisible by 9.
        So,the given number is divisible by 9.
(ii)Sum of digits in 36870521 is 32, which is not divisible by 9.
      So,the given is not divisible by 9.

Question:Divisiblity by 4:A number is divisible by 4,if the  number formed by the last two digits is divisible by 4.

8.Which of the following numbers is divisible by 4.?
  (i) 67920594  (ii) 618703572 

Sol.(i) The number formed by the last 2 digits in the given number is 94,
which is not divisible by 4.
 :. 67920594 is not divisible by 4.

(ii) The number formed by the last 2 digits in the given number is 72,which is divisible by 4.
:. 618703572 is divisible by 4.

Question:Divisiblity by 8: A number is divisible by 8,if the number formed by the last 3 digits of the given number is divisible by 8.

9.Which of the following numbers is divisible by 8.
  (i) 98016542  (ii) 106598304 

Sol.(i) The number formed by the last 3 digits of the given number is 542.
 which is not divisible by 8.
:. 98016542 is not divisible by 8.

(ii)The number formed by the last 3 digits of the given number is
 304, which is divisible by 8.
:. 106598304 is divisible by 8.

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. 

Sol.( 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.

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

Sol.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.

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

Sol.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)

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

Sol. `(i) 5793405 xx 99999 = 5793405 xx (100000 -1)`
                                 
                       = `(579340500000 - 5793405)`
                                  
                        =579334706595.

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

        (ii) `983 xx 207 - 983 xx 107` 

Sol. `(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.`

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)` 

Sol.(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

Question:16.Simplify :

   (i) `1605 xx 160 =?`

`(ii) 1398 xx 1398 =?` 

Sol. (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.