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

`24=3 xx 8` are co-prime.
The sum of the digits is 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 `3 xx 8` , i.e. `24`.

Question:Multiply `5793405` by `99999` by short cut method. 

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

Question:Evaluate: 
i) `896 xx 137 + 986 xx 863 ` 
ii) `983 xx 207 - 983 xx 107` 

i) `986 xx 137 + 986 xx 863` 
`= 986 ( 137 + 863 )`
`= 986 xx 1000`
`= 980000.`

ii) `983 xx 207 - 983 xx 107`
` = 983 xx ( 207 - 107)`
`= 983 xx 100 = 98300.`

Question:Evaluate following expressions:
 i) `( 527 xx 527 xx 527 + 183 xx 183 xx 183)/(527 xx 527 - 527 xx 183 + 183 xx 183)`
 
 ii) `(458 xx 458 xx 458 - 239 xx 239 xx 239)/(458 xx 458 xx + 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)` 

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) = (457 - 239) = 219)`


 iii) Given Expression

 `= ((a + b)^2 - (a - b)^2)/(ab)` 

 `= (4ab)/4`

 `= 4`


 iv) Given Expression

 `= ((a + b)^2 + (a - b)^2)/(a^2 + b^2)`

 `= (2(a^2 + b^2))/(a^2 + b^2)`

 `= 2`

Question:On dividing `15968` by a certain number, the quotient is `89` and the remainder is `37`. Find the divisor. 

Divisor = `(text{Dividend - Remainder})/text{Quotient} = (15968 - 37)/89 = 179`

Question:Convert the following into vulgar fractions:
`i) 0.75` 

 `ii) 3.004`

 `iii) .0056` 

i) `.075`

  `= 75/100`

 ` = 3/4`

 ii) `3.004` 

  `= 3004/1000`

 ` = 751/250`

 iii) `.0056/10000` 

    `= 7/1250`

Question:Simplify the following fractions:
i) `1.84/2.99` ii) `.365/.584` 

i) `1.84/2.99` = `184/299` = `8/13`
ii) `.365/.584` = `365/584` = `5/8`

Question:The H.C.F of two numbers is 11 and their L.C.M is 693. If one of the numbers is 77, find the other. 

The other number = `((11xx693)/77) = 99`

Question:Find the largest number that can exactly divide `513, 713` and `1107`. 

Required number 
= H.C.F. of `513, 783` and `1107`.
Now, `513 = 3^3 xx 19,` 
` 783 = 3^3 xx 29,` and 
`1107 = 3^3 xx 41`
`:.` H.C.F. = `3^3 = 27`
Hence, the required number is `27`

Question:Find the greatest number which can divide 284, 698 and 1618 leaving the same remainder 8 in each case. 

Required number = 
H.C.F. of `(284 - 8), (698 - 8)` & `(1618 - 8)`
= H.C.F. of `276, 690` and `1610.`

Now, 
`276 = 2^2 xx 3 xx 23,`
`690 = 2 xx 3 xx 5 xx 23,` 
`1610 = 2 xx 5 xx 7 xx 23`

`:.` H.C.F. of `276, 690` & `1610` is `23`.
Hence, the required number is `23`.