17. The difference of the squares of two different real numbers is param 1 times greater than the difference of these numbers, and the difference of the cubes of these numbers is param 2 times greater than the difference of these numbers. By how many times is the difference of the fourth powers of these numbers greater than the difference of the squares of these numbers?

The difference between the squares of two different real numbers is param 1 times greater than the difference between these numbers. The difference between the cubes of these numbers is param2 times greater than the difference between these numbers. Find the ratio of difference between the fourth powers of these numbers to the difference between squares of these numbers.

| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 37 | 1069 | 769 |
| 40 | 1209 | 818 |
| 37 | 1039 | 709 |
| 40 | 1201 | 802 |
| 31 | 741 | 521 |

## Condition

The difference of the squares of two different real numbers is 37 times greater than the difference of these numbers, and the difference of the cubes of these numbers is 1069 times greater than the difference of these numbers. By how many times is the difference of the fourth powers of these numbers greater than the difference of the squares of these numbers?

## Answer.

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# Solution

From the equalities $a^{2}-b^{2}=k(a-b)$ and $a^{3}-b^{3}=m(a-b)$ (in the problem $k=37, m=1069$), it follows that $a+b=k$ and $a^{2}+a b+b^{2}=m$. Squaring the first equality and subtracting the second from it, we get: $a b=k^{2}-m$. Then the ratio $\left(a^{4}-b^{4}\right):\left(a^{2}-b^{2}\right)=a^{2}+b^{2}=m-a b=2 m-k^{2}=769$.