Here we are dealing with two numbers which give the same remainder by 5 and by 7. It's useful to know that if you were to list all such numbers, you would get an equally spaced list, where the numbers are separated by the LCM of 5 and 7, so by 35. So x-y must be divisible by 35 here.
Of course, you could come up with sample numbers if you weren't familiar with the underlying theory. We need two numbers which give a remainder of 3 when divided by 5, and a remainder of 4 when divided by 7. We can start by listing small numbers which give a remainder of 3 when divided by 5. This list is equally spaced, by 5, so it's straightforward to generate a long list quickly:
3, 8, 13, 18, 23, 28, 33, 38, 43, 48, 53, 58, 63, 68, ...
Now if you scan this list looking for numbers which give a remainder of 4 when divided by 7, you'll see that 18 and 53 both work. So it might be that x=53 and y=18, and their difference is 35, from which we also get answer E.
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