Just for fun!-XIII

Algebra Level 3

Evaluate ( ∑ i = 1 2007 i ) ( m o d 1000 ) \left(\sum _{ i=1 }^{ 2007 }{ i }\right) \pmod {1000}


The answer is 28.

Swapnil Das by Swapnil Das , 20, India

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3 solutions

Swapnil Das
Aug 19, 2015

∑ i = 1 2007 i = 2007 × 2008 2 = 2007 X 1004 ≡ 7 × 4 ≡ 28 m o d 1000 \sum _{ i=1 }^{ 2007 }{ i } = \frac { 2007\times 2008 }{ 2 } =2007X1004\equiv 7\times 4 \equiv 28\mod 1000

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= ∑ i = 1 2007 i =\sum _{ i=1 }^{ 2007 }{ i }

= 1 + 2 + 3 + … + 2006 + 2007 = 2015028 = 1+2+3+\ldots+2006+2007 =2015028

2015028 ≡ 1000 ≡ 28 (mod 1000) \displaystyle 2015028 \equiv 1000\equiv\boxed{28} \text{(mod 1000)}

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I have a marvelous solution for this problem, but this margin is too narrow to contain it.

Swapnil Das - 5 years, 9 months ago
Dev Sharma
Aug 19, 2015

It is same as 2015028 (mod100) = 28

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