New tutor @CubeyThePenguin is supplying a lot of useful responses; but far too often he isn't answering the question that is asked; and -- even worse -- he is often giving answers without showing HOW to get the answers, which is of no benefit at all to the student.
And in this case he has given a wrong answer, also without showing any work....
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Here is the problem....
ABCD
* 4
----
DCBA
(1) A must be either 1 or 2; otherwise the product would not be 4 digits.
(2) Since the PIN multiplied by 4 has last digit A, A must be even.
(3) (1) and (2) tell us A is 2.
2BCD
* 4
----
DCB2
(4) D multiplied by 4 gives last digit 2, so D must be either 3 or 8.
(5) D is the leading digit of the product of 2BCD times 4, so D must be 8.
2BC8
* 4
----
8CB2
(6) 4 times C8 gives B2; 4 times C is even, and since 8 times 4 is 32, B must be odd. So B is 1, 3, 5, 7, or 9.
(7) 4 times 2BC8 gives a product with leading digit 8.
(8) (6) and (7) tell us B must be 1.
21C8
* 4
----
8C12
(9) 21?? times 4 gives 8C??, so C must be at least 4.
(10) C8 times 4 gives final two digits 12; that means C is either 2 or 7.
(11) (9) and (10) tell us that C is 7.
2178
* 4
----
8712