6. Lev took two natural numbers, added their sum to their product, and as a result got 1000. What numbers could Lev have taken? Find all options.
Solution. If we denote the numbers of Lev as a and b, we get: a + b + ab = 1000. Adding one to both sides: 1 + a + b + ab = 1001, or (1 + a)(1 + b) = 7 · 11 · 13. Since a and b are natural numbers, 1 + a > 1 and 1 + b > 1. Therefore, one of the six variants must be true:

a) 1 + a = 7, 1 + b = 11 · 13, from which a = 6, b = 142;

b) 1 + a = 11, 1 + b = 7 · 13, from which a = 10, b = 90;

c) 1 + a = 13, 1 + b = 7 · 11, from which a = 12, b = 76;

and three more variants that are obtained by swapping a and b.

Answer: 6 and 142, 10 and 90, 12 and 76.

Criteria.

Progress. Finding one or two examples -1 point; finding all three examples -2 points.