Need to find four-figure number.

Hello (Hi)

I need to find a **four-figure number which can be divided by 7 and which can be written as a sum of square and cube of any natural number..**

My English is not perfect, but I hope you will understand what I mean.

I would be very glad if someone could help me.

Re: Need to find four-figure number.

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Re: Need to find four-figure number.

Re: Need to find four-figure number.

10^3 + 1^2 = 1001 which is divisable by 7 143 times.

Re: Need to find four-figure number.

By the way: the square and the cube has to be of the same natural number. For example 10^2 + 10^3

Re: Need to find four-figure number.

It looks to me like this is just a matter of **trying**! We are looking a number, x, such that $\displaystyle x^2+ x^3$ has four digits and is divisible by 7. Since 7 is prime, that means that x must be divisible by 7. $\displaystyle 7^3= 343$ so for $\displaystyle x^3$ to be four digits itself the other factor in $\displaystyle x^3$ must be between 3 and 30. The only cubes in that interval are $\displaystyle 1= 1^3$, $\displaystyle 8= 2^3$, and $\displaystyle 27= 3^3$ so the only possible values for x are 7(1)= 7, 7(2)= 14, and 7(3)= 21.

If x= 7, then $\displaystyle 7^2+ 7^3= 49+ 343= 392$ which does not have four digits. If x= 14, then $\displaystyle 14^2+ 14^3= 196+ 2744=2940= 7(420)$. That has four digits and so is a perfectly good answer. If x= 21, then $\displaystyle 21^2+ 21^3= 441+ 9261= 9702= 7(1386)$. Both 2940 and 9702 fit the requirements.

Re: Need to find four-figure number.

Ok then, 14^3 + 14 ^ 2 which divided by 7 is 420.. May I have a thanks?

Re: Need to find four-figure number.

**HallsofIvy** Interesting explanation. My answer was 13, because 13^2+13^3 is 2366 and 2366 : 7 = 338, but I had no idea how to explain that. I think your solution is enough good. I don't know how to find the answer in other ways.