Question 21

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?

Solution

The 5 digits which are used in the combination lock are 9,8,7,5 and 4.

The sum of first 3 digits and the last 3 digits are divisible by 3. Hence, we can say that the third digit is common in both the combinations.

The possible groups of 3 digits whose sum is divisible by 3 are (9,8,7),(9,8,4),(9,7,5),(9,5,4).

Out of these 4 possible groups, we have to select two groups which can be used as the first and last 3 digits.

The selection should be done in such a way that both the groups should only have one digit in common.

Example : If we select (9,8,7) and (9,8,4) and let the middle digit be 9, then we will have 8 in both the first 3 digits and the last 3 digits which is only possible for the 3rd digit which is occupied by 9 in this case. Hence, this case is not possible.

So the only possible combinations for the first and last 3 digits are :

Case 1: (9,8,7) and (9,5,4) where the 3rd digit will be 9.

Case 2: (9,8,4) and (9,7,5) where the 3rd digit will be 9.

Now, it is given that the sum of the last 4 digits is divisible by 4.

The possible cases of the above scenario are :

Case 3: (9,8,7,4) where the first digit will be 5.

Case 4: (8,7,5,4)where the first digit will be 9 which is not possible as 9 is fixed as the 3rd digit from Case 1 and 2.

Since, the first digit is 5 and the third digit is 9, the second digit will either be 4 or 7 and the last two digits will be either (8,7) or (8,4)

So, the possible combinations of the lock are 54987, 54978, 57984 and 57948.

Hence, the option which is definitely false is E as no possible combinations has 8 as its 2nd digit.

$$\therefore\ $$ The required answer is E.