Q. Amit has forgotten his 4-digit locker key. He remembers that all the digits are positive integers and are different from each other. Moreover, the fourth digit is the smallest and the maximum value of the first digit is 3. Also, he recalls that if he divides the second digit by the third digit, he gets the first digit. How many different combinations does Amit have to try for unlocking the locker?
- 2
- 1
- 4
- 5
- 3
EXPLANATION
The maximum value that the first digit can take is 3. The fourth digit is the smallest.
So the following are the possible different values that the first and fourth digits can take…
(Remember, 0 is neither positive nor negative)
| First Digit | Fourth Digit |
| 3 | 2 |
| 3 | 1 |
| 2 | 1 |
If he divides the second digit by the third digit, he gets the first digit.
Case 1) If the first digit is 3.
The second digit is 3 times the third digit.
The possible pairs are…
(3, 1), (6, 2), (9, 3)
Since all the digits are distinct, only (6,2) works and it works only if the fourth digit is 1.
Case 2) If the first digit is 2.
The second digit is 2 times the third digit.
The possible pairs are…
(2, 1), (4, 2), (6, 3), (8, 4)
Since all the digits are distinct, only (6, 3) and (8, 4) work.
| First Digit | Second Digit | Third Digit | Fourth Digit |
| 3 | 6 | 2 | 1 |
| 2 | 6 | 3 | 1 |
| 2 | 8 | 4 | 1 |
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