Wednesday, June 16, 2010

Challenging P6

Find the sum of the digits of the first 100 odd numbers.


First 100 odd numbers 1, 3, 5, ....., 199

Digits of first 100 numbers

   1 ,   3 ,   5 ,   7 ,   9 --> 25
 11 , 13 , 15 , 17 , 19 --> 25 + 1 x 5
 21 , 23 , 25 , 27 , 29 --> 25 + 2 x 5
...
 91 , 93 , 95 , 97 , 99 --> 25 + 9 x 5

101 , 103 , 105, 107 , 109 --> 25 + 1 x 5
111 , 113 , 115 , 117 , 119 --> 25 + 1 x 5 + 1 x 5
121 , 123 , 125 , 127 , 129 --> 25 + 1 x 5 + 2 x 5
....
191 , 193 , 195 , 197 , 199 --> 25 + 1 x 5 + 9 x 5

20 x 25 = 500

1 x 5 + 2 x 5 + .... + 9 x 5 = 5 x (1 + 2 + 9)
                                         = 5 x (1+9)/2 x 9
                                         = 225

225 x 2 = 450

10 x 1 x 5 = 50

500 + 450 + 50 = 1000

The sum of digits of the first 100 odd numbers is 1000.

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