The sum of the reciprocals of three distinct natural numbers is 31/30 . What is the least possible product of three such natural numbers?

This problem can be solved by using the concept of harmonic progression and the properties of natural numbers.

Step 1: The problem states that the sum of the reciprocals of three distinct natural numbers is 31/30. This can be written as:

1/a + 1/b + 1/c = 31/30

Step 2: To simplify the equation, we can multiply each term by abc, which gives us:

bc + ac + ab = 31abc/30

Step 3: Rearrange the equation to isolate abc:

abc = 30ab + 30ac + 30bc

Step 4: We are looking for the least possible product of three such natural numbers, so we want to minimize abc. This will occur when a, b, and c are as small as possible.

Step 5: By trial and error, we can find that the smallest possible values for a, b, and c that satisfy the equation are a=1, b=5, and c=6.

Step 6: Therefore, the least possible product of three such natural numbers is 156 = 30.