When the positive integer A is divided by 5 and 7, the

If I have a number n which when divided by 5 gives a remainder 3 and when divided by 7 gives a remainder 4, the number is of the form:
n = 5a + 3
n = 7b + 4
I will need to check for the smallest such number.
I put b = 1. n = 11. Is it of the form 5a + 3? No.
Put b = 2. n = 18. Is it of the form 5a + 3? Yes.

When 18 is divided by 5, it gives a remainder of 3. When it is divided by 7, it gives a remainder if 4.

Next such number will be 35 + 18 because 35 will be divisible by 5 as well as 7 and whatever is the remainder from 18, will still be the remainder
Next will be 35*2 + 18
and so on...
Difference between such numbers will be a multiple of 35 so your answer is 35.
Note: Actually, because of this reasoning, you just had to take the LCM. You didn't even need to find the first such number!

I have discussed this topic a little more in detail here: https://gmatclub.com/forum/good-problem-90442-20.html#p814507