Answer:

Given 
three  positive integers a1, a2,a3, in AP
a1*a2*a3*=1155\


=> a1+a2+a3=33 eq1
according to AP
2[a2]=a1+a3
2[a2]=33-a2    [from 1]
3[a2]=33
a2=11 eq2

a1*a2*a3=1155
a1*a3=1155/11
a1*a3=105. eq3

from eq 1 
a1+a2+a3=33
a1+a3=22
=>   a1=22-a3

a1*a3=105    
[22-a3] a3=105
22a3-[a3]square=105
[a3]square -22a3 +105=0

by factorizing we get 
a3=15 or 7 eq 4 

we have a3  and a2
 we can find  rremaining a1  
we get a1=7 or 15

Given Information:
- We are given three positive integers, a1, a2, and a3, which are in the set A.
- The sum of these three integers is 55.
- The product of these three integers is 1155.

To Find:
We need to find the three integers a1, a2, and a3 that satisfy the given conditions.

Solution:
Let's solve this problem step by step.

Step 1: Write the Equations:
We can write two equations based on the given information:
- a1 + a2 + a3 = 55 (Equation 1)
- a1 * a2 * a3 = 1155 (Equation 2)

Step 2: Solve Equation 1:
Since a1, a2, and a3 are positive integers, we need to find three positive integers whose sum is 55. Let's try to find a solution by trial and error:

- If we assume a1 = 1, then a2 + a3 = 54. But there are no two positive integers that sum up to 54.
- If we assume a1 = 2, then a2 + a3 = 53. But again, there are no two positive integers that sum up to 53.
- If we assume a1 = 3, then a2 + a3 = 52. We can find two positive integers that sum up to 52, which are 26 and 26.
- Similarly, if we assume a1 = 4, we get a2 + a3 = 51. We can find two positive integers that sum up to 51, which are 25 and 26.

Step 3: Find the Possible Combinations:
From the above analysis, we found two possible combinations:
- Combination 1: a1 = 3, a2 = 26, a3 = 26
- Combination 2: a1 = 4, a2 = 25, a3 = 26

Step 4: Check the Product of the Combinations:
Now, let's check if these combinations satisfy the second equation (Equation 2) as well:
- Combination 1: 3 * 26 * 26 = 2028, which is not equal to 1155.
- Combination 2: 4 * 25 * 26 = 2600, which is not equal to 1155.

Step 5: Continue the Search:
Since the given combinations do not satisfy the second equation, we need to continue our search for other possible combinations. Let's try a1 = 5 and find the values of a2 and a3:

- If we assume a1 = 5, then a2 + a3 = 50. We can find two positive integers that sum up to 50, which are 25 and 25.

Step 6: Check the Product again:
Now, let's check if this new combination satisfies the second equation (Equation 2):
- Combination 3: 5 * 25 * 25 = 3125, which is not equal to 1155.

Step 7: Conclusion:
After searching for possible combinations, we did not