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Number Theory sheet -1 attaced

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

good evening sorry for being late . i tried to solve problems from 1 to 14 and i was not able solve problem number 7,9

- tpsriram
**Posts:**39**Joined:**Wed Feb 27, 2013 8:43 am

QCS-3- Number Theory-1

Q5. Sum of digits of the number (100 )^ 33 - 25

100 ^33 = 10 ^ 66 ( This number must be written as 1 followed by 66 zeros. Thus this number is a 67 digit number.

The number can be written as : 100000000000000000000 .......00 ( 66 zeros)

Thus when we subtract 25 from the above number we get

(100000000000000000000 .......00) - 25 = 9999999999999........75

( This number must have 66 digits where the last two digits are 7 and 5 and the rest of the digits are 9's

Thus the sum of digits of the number is : 64 x 9 + 7 + 5 = 588

Q5. Sum of digits of the number (100 )^ 33 - 25

100 ^33 = 10 ^ 66 ( This number must be written as 1 followed by 66 zeros. Thus this number is a 67 digit number.

The number can be written as : 100000000000000000000 .......00 ( 66 zeros)

Thus when we subtract 25 from the above number we get

(100000000000000000000 .......00) - 25 = 9999999999999........75

( This number must have 66 digits where the last two digits are 7 and 5 and the rest of the digits are 9's

Thus the sum of digits of the number is : 64 x 9 + 7 + 5 = 588

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

QCS -3 - Number Theory -

Q6 Which is greater - 17^31 or 31^17

Solution

17 > 16 ==> 17^31 > 16^31 => 17^ 31 > (2^4)^31 ===> 17^31 > 2^124

Similarly

31< 32 ===> 31^17 < 32 ^ 17 ====> 31^17 < (2^5)^17 ======> 31^17 < 2^85

Thus from the above we can easily infer that 17^31 > 31^17

Q6 Which is greater - 17^31 or 31^17

Solution

17 > 16 ==> 17^31 > 16^31 => 17^ 31 > (2^4)^31 ===> 17^31 > 2^124

Similarly

31< 32 ===> 31^17 < 32 ^ 17 ====> 31^17 < (2^5)^17 ======> 31^17 < 2^85

Thus from the above we can easily infer that 17^31 > 31^17

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

QCS-3- Number Theory-1

Q9

Solution

(1-1/3)(1-1/4)(1-1/5) .........(1-1/n) = 2/3 * 3/4 * 4/5 .........(n-1)/n

= 2/n

Q9

Solution

(1-1/3)(1-1/4)(1-1/5) .........(1-1/n) = 2/3 * 3/4 * 4/5 .........(n-1)/n

= 2/n

- sadmin
- Site Admin
**Posts:**228**Joined:**Wed Feb 13, 2013 5:51 am

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