# Need to find four-figure number.

• Feb 19th 2013, 03:15 AM
Sott
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.
• Feb 19th 2013, 04:21 AM
agripa
Re: Need to find four-figure number.
H. Please write me personal in skype: agripasiga i'wiill try to help solve you problem
• Feb 19th 2013, 04:27 AM
Sott
Re: Need to find four-figure number.
You can write here.
• Feb 19th 2013, 04:51 AM
MathJack
Re: Need to find four-figure number.
10^3 + 1^2 = 1001 which is divisable by 7 143 times.
• Feb 19th 2013, 05:08 AM
Sott
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
• Feb 19th 2013, 05:11 AM
HallsofIvy
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 $x^2+ x^3$ has four digits and is divisible by 7. Since 7 is prime, that means that x must be divisible by 7. $7^3= 343$ so for $x^3$ to be four digits itself the other factor in $x^3$ must be between 3 and 30. The only cubes in that interval are $1= 1^3$, $8= 2^3$, and $27= 3^3$ so the only possible values for x are 7(1)= 7, 7(2)= 14, and 7(3)= 21.

If x= 7, then $7^2+ 7^3= 49+ 343= 392$ which does not have four digits. If x= 14, then $14^2+ 14^3= 196+ 2744=2940= 7(420)$. That has four digits and so is a perfectly good answer. If x= 21, then $21^2+ 21^3= 441+ 9261= 9702= 7(1386)$. Both 2940 and 9702 fit the requirements.
• Feb 19th 2013, 05:12 AM
MathJack
Re: Need to find four-figure number.
Ok then, 14^3 + 14 ^ 2 which divided by 7 is 420.. May I have a thanks?
• Feb 19th 2013, 05:29 AM
Sott
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.