Find the greatest common divisor
- To find the greatest number which divides 625 and 1433 leaving remainders 5 and 3 respectively, we need to find the greatest common divisor (GCD) of the two numbers.
- Let's denote the numbers as A = 625 and B = 1433, and the remainders as a = 5 and b = 3.

Calculate the differences
- First, we subtract the remainders from the numbers: A - a = 625 - 5 = 620 and B - b = 1433 - 3 = 1430.
- Now we find the GCD of 620 and 1430 using the Euclidean algorithm.

Apply the Euclidean algorithm
- Divide the larger number by the smaller number: 1430 ÷ 620 = 2 with a remainder of 190.
- Now, divide 620 by 190: 620 ÷ 190 = 3 with a remainder of 50.
- Continue this process until we reach a remainder of 0: 190 ÷ 50 = 3 with a remainder of 40, 50 ÷ 40 = 1 with a remainder of 10, and 40 ÷ 10 = 4 with a remainder of 0.

Identify the greatest common divisor
- The last non-zero remainder is 10, which means that the greatest common divisor of 620 and 1430 is 10.
- Therefore, the greatest number that divides 625 and 1433 leaving remainders 5 and 3 respectively is 10.