Question

what is the reminder when 1^3+2^3+3^3+……..+1000^3 is divided by 13
8​

Answer 1

Step-by-step explanation:

sum of cubes of n natural numbers = (n×n+1 /2)²

sum of cubes of 1000 natural no.s =( 1000×1001/2)²

its exactly divisible by 13

it can be written as( (1001-1)*1001/2)²

=(( (77×13)-1)×13×77/2)²

let 13 be x

f(x)=((77x-1)×x×7/2)² /x

when f(x) is divided by x then remainder is f(0)

f(0) = 0

Answer 2

Sum of cubes of n natural no. = $\frac{(n*n+1)^{2} }{2}$

sum of cubes of 1000 natural no.  = $(1000*\frac{1001}{2} )^{2}$

this is exactly divisible by 13

it can be written as $((1001-1)$×$(\frac{1001}{2} ))^{2}$

=(( (77×13)-1)×13×77/2)²

when f(x) is divided by x then remainder is f(0)

f(0) = 0

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