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?

• A. 2
• B. 1
• C. 4
• D. 5
• E. 3

Answer 1

Answer: (E)

Amit needs a 4-digit locker code where all digits are different positive integers. Let's break down the conditions:

1. The digits are positive integers with a maximum value of 9.
2. All digits are distinct.
3. The first digit has a maximum value of 3.
4. The fourth digit is the smallest.
5. The second digit divided by the third digit equals the first digit.

Given the constraints, let's denote the digits as \(a, b, c,\) and \(d\) where: \(a\) is the first digit, \(b\) is the second digit, \(c\) is the third digit, \(d\) is the fourth digit. According to the problem, \(a = \frac{b}{c}\) and the smallest number \(d\) should take the smallest available value that satisfies all conditions:

1. The possible values of \(a\) are 1, 2, or 3.
2. The smallest possible value for \(d\) is 1.

We need to try different permutations for \(a = 1, 2, 3\):

• For \(a = 3\):
  • \(b = 6, 9\), \(c = 2, 3\)
  • Possible 4th digit is 1 (the smallest available).
• For \(a = 2\):
  • \(b = 2, 4, 6, 8\), \(c = 1, 2, 3, 4\)
• For \(a = 1\):
  • \(b = 1, 2, 3, 4, 5, 6, 7, 8, 9\), \(c = b\)

Counting valid sequences from these choices:

• Combination with \(a = 3\), \(b = 9, c = 3, d = 1\) is possible.
• Combination with \(a = 3\), \(b = 6, c = 2, d = 1\) is possible.
• Combination with \(a = 2\), \(b = 4, c = 2, d = 1\) is possible.

Therefore, the total possible unique combinations are:

3 solutions.