Split 597 into three parts such that these are in A.P. and the product of the two smallest

parts is 796.

To solve this problem, we can start by assuming the three parts of the given number, 597, are represented as (a - d), a, and (a + d), where 'a' is the middle term and 'd' is the common difference in the arithmetic progression (A.P.).

Now we can form the equations based on the given information:

1) (a - d) * a = 796 (The product of the two smallest parts is 796)
2) (a - d) + a + (a + d) = 597 (The sum of the three parts is 597)

Let's solve these equations step by step:

1) (a - d) * a = 796

Expanding the equation, we get:
a^2 - ad = 796

2) (a - d) + a + (a + d) = 597

Simplifying the equation, we get:
3a = 597

Now, we can solve these two equations simultaneously.

From equation (2), we can find the value of 'a':
3a = 597
a = 597 / 3
a = 199

Substituting this value of 'a' in equation (1), we get:
199^2 - 199d = 796
(dropping the squared term)

199 - d = 4
d = 199 - 4
d = 195

Now we have the value of 'a' and 'd', we can find the three parts:

Part 1: (a - d) = 199 - 195 = 4
Part 2: a = 199
Part 3: (a + d) = 199 + 195 = 394

So, the three parts of 597 that are in an arithmetic progression (A.P.) and satisfy the condition of the product of the two smallest parts being 796 are 4, 199, and 394.