Question

Ronny uses a 5-digit key for a combination lock, where 5 digits need to be entered in a fixed sequence. While he remembers that the 5 digits are 9, 8, 7, 5 and 4, he has forgotten the sequence he uses. He also remembers that the sum of the first three digits is a multiple of 3, and so is the sum of the last three digits. Further, the sum of the last four digits is a multiple of 4.

Which of the following is DEFINITELY FALSE?

Accepted Answer

E. Digits available: $\{9, 8, 7, 5, 4\}$. Sum of first 3 digits and sum of last 3 digits are divisible by 3 $\implies$ the 3rd digit is the common middle digit. Subsets of 3 digits with sum divisible by 3: $(9,8,7), (9,8,4), (9,7,5), (9,5,4)$. To share only 1 middle digit, 9 must be the 3rd digit. First/Last sets are $(9,8,7)$ and $(9,5,4)$ or $(9,8,4)$ and $(9,7,5)$. Sum of last 4 digits divisible by 4: Possible valid 5-digit sequences satisfying all conditions are: 54987, 54978, 57984, 57948. In all valid combinations, the second digit is either 4 or 7. It can never be 8. Thus, "8 is the second digit of the key" is DEFINITELY FALSE.

Combination lock 5-digit key

Explanation

Practise this under exam conditions