B3. Computationally determine all possible pairs of natural numbers $a$ and $b, a>b$, such that the difference of the squares of these two numbers is equal to 60.

## 18th Knowledge Competition

 mathematics for students of secondary technical and vocational schoolsNational Competition, April 21, 2018

## Problems for 2nd Year

Time for solving: 120 minutes. In section A, we will award three points for each correct answer, and deduct one point for each incorrect answer. Write the answers for section A in the left table, leave the right table blank.

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-06.jpg?height=163&width=319&top_left_y=661&top_left_x=640)

| B1 | B2 | B3 |
| :--- | :--- | :--- |
|  |  |  |
B3.
Expression $a^{2}-b^{2}=60$ ..... 1 point
Factoring the difference of squares ..... 1 point
Listing all possible factorizations ..... 1 point
Writing at least one system of equations ..... 1 point
Solving the system of equations ..... $1^{*}$ point
Determining that for the factorizations $1 \cdot 60, 3 \cdot 20, 4 \cdot 15, 5 \cdot 12$ the solutions are not natural numbers ..... 1 point
Solution $a=16$ and $b=14$ for the factorization $60=2 \cdot 30$ ..... 1 point
Solution $a=8$ and $b=2$ for the factorization $60=6 \cdot 10$ ..... 1 point

## 18th Mathematics Knowledge Competition for Students of Secondary Technical and Vocational Schools  National Competition, April 21, 2018

## Solutions and Scoring Guide for Second Year

A competitor who arrives at the solution by any correct method (even if not anticipated by the scoring guide) receives all possible points.

A correct method is one that:

- sensibly takes into account the wording of the problem,
- leads to the solution of the problem,
- is mathematically correct and complete.

If an intermediate or final result can be recognized, guessed, read from a diagram, or calculated mentally, the competitor generally receives all the points provided. If, however, the solution is guessed (cannot be reached without calculation), even if it is correct, it is scored with 0 points.

A competitor who has only partially solved the problem, with correct procedures but without a clear path to the final solution, cannot receive more than half of the possible points.

The '*' mark next to the points means that the point or points can be awarded for a correct procedure, even if the calculation is incorrect.

![](https://cdn.mathpix.com/cropped/2024_06_07_70800c8f4239f8164bacg-20.jpg?height=140&width=217&top_left_y=1552&top_left_x=931)