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Find the last three digits of the sum of the first 2014 whole numbers.

105 078 None of the rest 091

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1 solution

Prasun Biswas
Feb 23, 2014

The set of whole numbers starts from 0 and consists of all positive integers.

So, the series of the first 2014 whole nos. is : 0 , 1 , 2 , 3 , . . . . . , 2013 0,1,2,3,.....,2013 . Now, we use here the identity 1 + 2 + 3 + . . . . + n = n ( n + 1 ) 2 1+2+3+....+n=\frac{n(n+1)}{2}

0 + 1 + 2 + 3 + . . . + 2013 = 1 + 2 + 3 + . . . . + 2013 = 2013 × 2014 2 = 2013 × 1007 = 2027091 0+1+2+3+...+2013 = 1+2+3+....+2013 = \frac{2013\times 2014}{2}=2013\times 1007 = \boxed{2027091}

So, the last three digits of the sum is = 091 =\boxed{091}

Also, you can find the last three digits of the sum by multiplying the numbers formed by the last three digits of 1007 1007 and 2013 2013 , i.e, by multiplying the numbers 007 007 and 013 013 and checking the last three digits of that product.

007 × 013 = 7 × 13 = 91 = 091 007\times 013 = 7\times 13 = 91 = 091

So, the last three digits of that product is 091 091 , so the last three digits of the original product is also 091 \boxed{091}

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