6.2. Second-graders Kolya, Vasya, Misha, Stepa, and Grisha took turns correctly solving five multiplication table problems. Each subsequent boy received an answer one and a half times greater than the previous one. What numbers did Stepa multiply?
Answer: 6 and 9.

Solution. First method. Each boy received an answer $\frac{3}{2}$ times greater than the previous one. Therefore, Grisha received an answer $\frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2} \cdot \frac{3}{2}=\frac{81}{16}$ times greater than Kolya. Since all the numbers received by the boys are integers, Grisha's number must be a multiple of 81. But 81 is the largest number in the multiplication table, so Grisha received exactly 81. Then Stepa received $81: \frac{3}{2}=54$. The number 54 can be uniquely represented as the product of two single-digit numbers (from the multiplication table): $54=6 \times 9$.

Second method. Let Kolya receive the number $x$ in the answer. Then Vasya received $x \cdot 1.5=\frac{3 x}{2}$, Misha $-\frac{3 \cdot 3 x}{2 \cdot 2}$, Stepa $-\frac{3 \cdot 3 \cdot 3 x}{2 \cdot 2 \cdot 2}$, and Grisha $-\frac{3 \cdot 3 \cdot 3 \cdot 3 x}{2 \cdot 2 \cdot 2 \cdot 2}$. Grisha's number is an integer, and the numbers 2 and 3 are coprime, so the factorization of $x$ into prime factors must include at least four twos.

If there are no other factors in this factorization, then Kolya received 16, Vasya - 24, Misha - 36, Stepa - 54, and Grisha - 81. In this case, Stepa multiplied the numbers 6 and 9.

Suppose there is something else in this factorization besides the four twos, then this "something" would multiply Grisha's result of 81, but then it would not fit into the multiplication table. Therefore, other cases are impossible.

## Evaluation criteria:

+ correct answer and solution with full justification provided

$\pm$ only the correct multiplication result for Stepa is obtained with justification, but the factors are not found

干 correct multiplication results for someone other than Stepa are found with justification

$\mp$ correct answer and verification that it meets the condition are provided, but it is not justified that it is the only one

- only the answer is provided