199 × the quotient equals the number we’re seeking (7777 …).

Hence (200 × the quotient) minus the quotient will give us the same number.

So let a, b, c, and d be the last four digits of the quotient. Now the quotient multiplied by 200 ends in r, s, 0, 0, where r, s are the digits that result from multiplying c, d by 2.

Here’s the subtraction:

So d must be 3, and c must be 2.

Therefore s, which we know is twice d, is 6, and r, which is twice c, is 4. And that tells us that b is 8 and a is 6.

So the last four digits of the quotient are 6823.