Task 3. Sasha chose five numbers from the numbers 1, 2, 3, 4, 5, 6, and 7 and told Anna their product. Based on this information, Anna realized that she could not uniquely determine the parity of the sum of the numbers chosen by Sasha. What number did Sasha tell Anna? (20 points)
Answer: 420.

Solution: Let's look at the two remaining numbers. Since Anya knows the sum of all numbers from 1 to 7 (which is 28), these two remaining numbers are such that their product cannot determine the parity of their sum. Therefore, their product can be represented in two ways: \( ab = xy \), where numbers \( a \) and \( b \) are of different parity, and numbers \( x \) and \( y \) are of the same parity. Then \( ab \) is even, so \( xy \) is even, which means one of them is divisible by 2, but then \( x \) and \( y \) are both even. Therefore, \( xy = ab \) is divisible by 4, and thus the even number among \( a \) and \( b \) is 4 (Anya simply has no other multiples of four). Then the pair of numbers \( x \) and \( y \) must be 2 and 6. Their product is 12, so the number paired with 4 is 3.

Thus, the product of the two remaining numbers is necessarily 12. Therefore, the product of Sasha's numbers is \( 1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7 / 12 = 420 \).

Criteria: Only the answer - 5 points.