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

parts is 796.

let 597=x+y+z(given)

Also 796=xy(given)
And y-x=z-y (bcs in A.P.)
So z=2y-x y=796/x
z=1592/x-x
X+796/x+1592/x-x=597
X =4
y =796/4=199
z =1592/4-4=394

To split 597 into three parts that are in an arithmetic progression (A.P.), let's assume the first term of the A.P. is a and the common difference is d.

So, the three parts in A.P. can be written as:
First term: a
Second term: a + d
Third term: a + 2d

Given that the product of the two smallest parts is 796, we can set up the equation:
(a)(a + d) = 796

Expanding the equation, we get:
a^2 + ad - 796 = 0

Now, we need to factorize this quadratic equation using the method of decomposition. We need to find two numbers whose product is the constant term (-796) and whose sum is the coefficient of the middle term (1, as the coefficient of 'ad' is 1).

After some calculations, we find that the numbers are 28 and -29. Therefore, we can rewrite the equation as:
(a + 28)(a - 29) = 0

Setting each factor equal to zero, we can solve for the value of 'a':
a + 28 = 0 --> a = -28
a - 29 = 0 --> a = 29

Since the terms of the A.P. cannot be negative, we can discard -28 as a valid solution. Therefore, the first term (a) of the A.P. is 29.

Now, we can find the common difference (d) by subtracting the first term from the second term:
Second term - First term = (a + d) - a
d = 2d - d = a + d - a = 29

Therefore, the common difference (d) is 29.

The three parts in A.P. are:
First term: a = 29
Second term: a + d = 29 + 29 = 58
Third term: a + 2d = 29 + 2(29) = 87

So, the three parts of 597 in an A.P. such that the product of the two smallest parts is 796 are 29, 58, and 87.