Two numbers are such that the sum of their cubes is 5 and the sum of their squares is 3. Find the sum of the numbers.

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- Oct 11th 2008, 06:48 AMgreat_mathA good question
Two numbers are such that the sum of their cubes is 5 and the sum of their squares is 3. Find the sum of the numbers.

- Oct 11th 2008, 07:03 AMMoo
Hello,

Note that

If a and b are integers, then there are a few possibilities :

3-ab=1 and a+b=5 (1)

3-ab=5 and a+b=1 (2)

3-ab=-1 and a+b=-5 (3)

3-ab=-5 and a+b=-1 (4)

From (1), we get ab=2, that is a=1 and b=2 for example. But a+b is not equal to 5. Impossible for integers.

From (2), we get ab=-2. We want a+b=1. --> a=-1 and b=2.

From (3), we get ab=4 --> a=-1 and b=-4

From (4), we get ab=8. Impossible for integers. - Oct 11th 2008, 07:36 AMJhevon
Alternatively, let the numbers be and . we have

So as Moo said,

that is,

Now,

which means

Now, let and , then we have the system

.....................(1)

....................(2)

Solve this system for and you have your answer (since is the sum of the two numbers)

I like your solution, Moo :D

But, of course, who said they had to be integers? - Oct 11th 2008, 07:40 AMMoo
- Oct 15th 2008, 08:57 AMsharon333
Thanks for the formulas.