A, B and C are different prime numbers, and positive integer D is a no

HI
you are correct. I did check for A as 3 and B as 2 with C as 5 and 7, but missed out checking for A=5.
Kudos

We know that

D= B * C + A
A<C

take B=3; C=5 and A=3
--> 18 = 3 * 5 + 3 --> min(B +C) = 8

All restrictions are satisfied, is there anything wrong in my reasoning?

Good idea!
The problem is that the question says "A, B and C are different prime numbers"

Why cant A be 7 when I take B as 3 and C as 5?
15 + 7 gives me 22 which is not prime

Posted from my mobile device

A has to be less than C.

D divide by C or 5 cannot give A or 7 as remainder. Remainder has to be less than the divisor or 5 here.

Given: A, B and C are different prime numbers, and positive integer D is a non-prime number.
Asked: If D divided by C equals B with remainder A, what is the smallest possible value of B + C?

D = BC + A

Prime numbers = {2,3,5,7,11,13,17,19,23,29,}

For the smallest possible value of B+C, BC = 2*3 = 6; B+C=5; No value of A is possible; Rejected
For next smallest possible value of B+C, BC = 2*5 = 10; B+C=7; A=3; D= 13; Rejected
For next smallest possible value of B+C, BC = 3*5 = 15; B+C=8; A=2; D=17; Prime; Rejected
For next smallest possible value of B+C, BC = 2*7 = 14; B+C=9; A=3; D=17; Prime; Rejected; A=5;D=19;Prime; Rejected
For next smallest possible value of B+C, BC = 3*7 = 21; B+C=10; A=2; D=23; Prime; Rejected; A=5;D=26=2*13; Non-prime: Accepted

26 = 3*7 + 5
D = 26; C=7; B=3; A=5; B+C = 3+7 = 10

​​​​​​​IMO B­