Question

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?

Accepted Answer

E. Let digits be $a, b, c, d$. All positive ($1-9$), distinct, $d < a, b, c$, $a \le 3$, and $b/c = a \implies b = ac$. Since $a \le 3$ and $a > d$, $a$ cannot be 1. Thus $a=2$ or $a=3$. Case 1: $a=2 \implies d=1$. $b = 2c$. If $c=3 \implies b=6$, code is $2631$. If $c=4 \implies b=8$, code is $2841$. Case 2: $a=3 \implies d=1$ or $d=2$. $b = 3c$. If $d=1 \implies c=2 \implies b=6$, code is $3621$. If $d=2$, no integer $c$ works with distinct non-repeating digits. Total combinations = 3.

4-Digit Locker Key Combination

Explanation

Practise this under exam conditions