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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?

Explanation:

Let the number be 'abcd' where all digits are positive single-digit distinct integers.

Now, d is the smallest of all the 4 digits while a = b/c.
Also, a ≤ 3.

Case 1: a = 1
Not possible since d is the smallest of all the digits.

Case 2: a = 2 ⇒ b = 2c
Also, since d is the smallest of the 4 digits, d = 1
Now, possible values of (b, c) are (2, 1), (4, 2), (6, 3) and (8, 4)
But c cannot be the least of the 4 digits and all digits must be distinct.
∴ Accepted values of (b, c) are (6, 3) and (8, 4)
⇒ (a, b, c, d) can be (2, 6, 3, 1) or (2, 8, 4, 1)

Case 3: a = 3 ⇒ b = 3c
Also, since d is the smallest of the 4 digits, d = 1 or 2
Now, possible values of (b, c) are (3, 1), (6, 2), (9, 3)
But c cannot be the least of the 4 digits and all digits must be distinct.
∴ Accepted values of (b, c) are (6, 2)
⇒ (a, b, c, d) can be (3, 6, 2, 1)

∴ Total acceptable values of (a, b, c, d) are (2, 6, 3, 1) or (2, 8, 4, 1) or (3, 6, 2, 1) i.e., 3 values.

Hence, option (e).

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