If , find the value of the 4-digit integer , ( ) and are distinct).
(Source: SASMO Primary)
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Note the (original) problem didn't require A , B , C , D to be distinct; however, we don't need this to find a solution.
A B C D can't be more than 1 1 1 1 , otherwise 9 × A B C D would have five digits. So A = 1 .
Since 1 0 0 0 ≤ A B C D ≤ 1 1 1 1 , we have 9 0 0 0 ≤ 9 × A B C D ≤ 9 9 9 9 and D = 9 .
Now we have 9 × 1 B C 9 = 9 C B 1 , so 9 × ( 1 0 0 0 + 1 0 0 B + 1 0 C + 9 ) 9 0 8 1 + 9 0 0 B + 9 0 C 8 9 B + 8 = 9 0 0 0 + 1 0 0 C + 1 0 B + 1 = 9 0 0 1 + 1 0 0 C + 1 0 B = C
From this last line, it's clear we can't have B > 0 , so B = 0 , C = 8 and A B C D = 1 0 8 9 .