Does there exist an integer, the product of whose digits is a) 1980? b) 1990? c) 2000?
This task can be reformulated as: "Can the numbers 1980, 1990, 2000 be factored into single-digit factors?"

## Solution

Among the divisors of the number 1980 is a two-digit prime number 11, and among the divisors of the number 1990 is a three-digit prime number 199, so neither 1980 nor 1990 can be factored into such products. The number 2000 can be factored into single-digit factors in various ways, for example: \(2000 = 2^4 \times 5^3\). Therefore, there are many such numbers as required by the condition. For example, two of them are 555422 and 25855.

## Answer

a-b) Does not exist; c) exists.