Finding the Four-Digit Pin

The problem describes a four-digit pin, say \(abcd\), where \(a\), \(b\), \(c\), and \(d\) are distinct non-zero digits. This means each digit must be one of \(1, 2, 3, 4, 5, 6, 7, 8, 9\), and all four digits must be different from each other.

We are given the following relationships between the digits:

• \(b = 2a\)
• \(c = 2b\)
• \(d = 2c\)

We can substitute the first equation into the second, and the second into the third, to express all digits in terms of \(a\):

• \(b = 2a\)
• \(c = 2b = 2(2a) = 4a\)
• \(d = 2c = 2(4a) = 8a\)

Now, we need to find a value for \(a\) such that \(a\), \(b=2a\), \(c=4a\), and \(d=8a\) are all distinct non-zero digits (between 1 and 9 inclusive).

Let's try possible non-zero values for \(a\):

• If \(a = 1\):
  • \(b = 2 \times 1 = 2\)
  • \(c = 4 \times 1 = 4\)
  • \(d = 8 \times 1 = 8\)
  The digits are 1, 2, 4, and 8. These are all distinct and non-zero. This is a valid set of digits. The pin is 1248.
• If \(a = 2\):
  • \(b = 2 \times 2 = 4\)
  • \(c = 4 \times 2 = 8\)
  • \(d = 8 \times 2 = 16\)
  The digit \(d\) is 16, which is not a single digit. This is not a valid pin.

If we try any value of \(a\) greater than 1, the value of \(d=8a\) will be 16 or larger, which is not a single digit. Therefore, the only possible value for \(a\) is 1.

The unique four-digit pin that satisfies all the conditions is 1248.

Checking Divisibility of the Pin 1248

The question asks by which numbers the pin 1248 is divisible among the given options. The options involve checking divisibility by 2, 3, and a third number (5, 7, 13, or 11).

Divisibility by 2

A number is divisible by 2 if its last digit is an even number (0, 2, 4, 6, or 8).

The pin is 1248. The last digit is 8, which is an even number.

Therefore, 1248 is divisible by 2.

Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3.

The sum of the digits of 1248 is \(1 + 2 + 4 + 8 = 15\).

The number 15 is divisible by 3 (\(15 = 3 \times 5\)).

Therefore, 1248 is divisible by 3.

Checking the Third Divisor from Options

Since 1248 is divisible by both 2 and 3, we now check the third number provided in each option.

Divisibility by 5 (Option 1: 2, 3, 5)

A number is divisible by 5 if its last digit is 0 or 5.

The last digit of 1248 is 8. It is not 0 or 5.

Therefore, 1248 is not divisible by 5.

Divisibility by 7 (Option 2: 2, 3, 7)

To check divisibility by 7, we can subtract twice the last digit from the number formed by the remaining digits. We repeat this process until we get a small number.

For 1248: Number is 124, last digit is 8. Calculate \(124 - 2 \times 8 = 124 - 16 = 108\).

For 108: Number is 10, last digit is 8. Calculate \(10 - 2 \times 8 = 10 - 16 = -6\).

Since -6 is not divisible by 7, 1248 is not divisible by 7.

Divisibility by 13 (Option 3: 2, 3, 13)

We can perform division to check if 1248 is divisible by 13.

Divide 1248 by 13:

\(1248 \div 13\)

\(13 \times 90 = 1170\)

\(1248 - 1170 = 78\)

\(13 \times 6 = 78\)

\(78 - 78 = 0\)

Since the remainder is 0, 1248 is divisible by 13 (\(1248 = 13 \times 96\)).

Thus, 1248 is divisible by 2, 3, and 13. This matches Option 3.

Divisibility by 11 (Option 4: 2, 3, 11)

To check divisibility by 11, find the alternating sum of the digits, starting from the rightmost digit.

For 1248: \(8 - 4 + 2 - 1 = 5\).

Since the alternating sum (5) is not 0 or a multiple of 11, 1248 is not divisible by 11.

Conclusion

The unique four-digit pin satisfying the given conditions is 1248. This pin is divisible by 2, 3, and 13.

Revision Table: Pin Calculation and Divisibility Checks

Additional Information: Number Theory Concepts

This problem involves basic concepts from number theory, specifically properties of integers and divisibility rules. Understanding these rules helps in quickly determining if one integer can be exactly divided by another without performing long division.

Properties of Digits and Numbers

A number is composed of digits. The place value of each digit is crucial. For a number like \(abcd\), its value is \(1000a + 100b + 10c + d\). The problem constraints on digits (distinct, non-zero) limit the possibilities significantly.

Common Divisibility Rules

Divisibility rules are shortcuts. Some common rules include:

• Divisibility by 2: A number is divisible by 2 if its unit digit is 0, 2, 4, 6, or 8.
• Divisibility by 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
• Divisibility by 4: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
• Divisibility by 5: A number is divisible by 5 if its unit digit is 0 or 5.
• Divisibility by 6: A number is divisible by 6 if it is divisible by both 2 and 3.
• Divisibility by 9: A number is divisible by 9 if the sum of its digits is divisible by 9.
• Divisibility by 10: A number is divisible by 10 if its unit digit is 0.
• Divisibility by 11: A number is divisible by 11 if the alternating sum of its digits (starting from the rightmost digit) is 0 or a multiple of 11.

Checking divisibility by prime numbers like 7, 13, 17, etc., sometimes involves more complex rules or direct division, especially for larger numbers.