X^3+y^3+z^3+k^3

[eldar]

If x = (16^3 + 17^3 + 18^3 + 19^3), then x divided by 70 leaves a remainder of

A = 0
B = 1
C = 3
D = 35
E = 69

[Lock]

I think the remainder here is 0. My reasoning is as follows:
The denominator (70) can be written in the form (16+17+18+19). Since these same elements (and no others) are 'mirrorred' in the numerator and raised to the same power, we can say that numerator can be divided by denominator without leaving any decimal digit/s. Hence, A.

[Lock]

This should be added to the above statement: "without leaving any decimal digit/s EXCEPT zero".

[eldar]

Hi can you explain how to have proof of your formula?

I tried on excel and while for the original question worked here is is what i found when using different numbers:

16,17,10,19-->16^3=4096,17^3=4913, 10^3=1000 and 19^3=6859
sum of cubes is 16868, sum of numbers is 62.

But 16868/62=272.0645

If my computer is still working...is the formula working only for certain numbers?

[Lock]

Hi can you explain how to have proof of your formula?

I tried on excel and while for the original question worked here is is what i found when using different numbers:

16,17,10,19-->16^3=4096,17^3=4913, 10^3=1000 and 19^3=6859
sum of cubes is 16868, sum of numbers is 62.

But 16868/62=272.0645

If my computer is still working...is the formula working only for certain numbers?

I guess, the numbers should be consecutive integers. Hope this helps!

[eldar]

I have found the genral formula->

sum of cubes of consecutive integers= square of their sum

very nice

[nithesh]

I have found the genral formula->

sum of cubes of consecutive integers= square of their sum

very nice

How did you derive this?

[nithesh]

(a+b)3 = a3+3ab(a+b)+b3
=> a3 + b3 = (a+b)3 - 3ab(a+b)

16^3 + 19^3 = (16+19)^3 - 3*16*19*(16+19)
{As 16+19=35}
= 35^3 - 35*(3*16*19)

llly......
17^3 + 18^3 = 35^3 - 35*(3*17*18)

Hence,
16^3 + 17^3 + ^18^3 + 19^3
= 35^3 - 35*(3*16*19) + 35^3 - 35*(3*17*18)
= 2*35^3 - 35*(EVEN) - 35*(EVEN)

Every part of this expression is a multiple of 35 & an even number

That means when divided by 70 the remainder will be ZERO

[nithesh]

I think the remainder here is 0. My reasoning is as follows:
The denominator (70) can be written in the form (16+17+18+19). Since these same elements (and no others) are 'mirrorred' in the numerator and raised to the same power, we can say that numerator can be divided by denominator without leaving any decimal digit/s. Hence, A.

Do you think this logic would work in case of 12,13,14,15,16??

[nithesh]

Simplest answer to this would be:

x is even and when divided by an even number (70), the remainder has to be an even number too.
As all the other 4 options are odd numbers, zero is the answer!!!