Find The Missing Digits: A Tricky Password Puzzle

by SLV Team 50 views

Hey guys, let's dive into a cool math problem that involves figuring out Gülcan's phone password. She's picked a four-digit code that starts with '1', has a mystery digit, then a '2', and another mystery digit. But here's the kicker – this password plays by some pretty specific rules. It's divisible by 5 without any remainder, and when you divide it by 9, you're left with a remainder of 8. Our mission, should we choose to accept it, is to find out what those missing digits could be and then figure out the possible sums of those digits. Buckle up; it's gonna be a fun ride!

Cracking the Code: The Divisibility Rules

Okay, so first things first, let's talk about those divisibility rules because they're going to be our best friends in cracking this code. A number is divisible by 5 if its last digit is either a 0 or a 5. That's the golden rule right there. Now, when it comes to divisibility by 9, things get a tad more interesting. A number is divisible by 9 if the sum of its digits is divisible by 9. But, in our case, we're not looking for perfect divisibility; we want a remainder of 8. So, the sum of the digits should be 8 more than a multiple of 9. Got it? Great, let's move on!

Applying the Rules to Gülcan's Password

Gülcan's password looks like this: 1 * 2 ▲. Now, let's use our divisibility rule for 5. The last digit (▲) must be either 0 or 5. This gives us two possible password formats: 1 * 20 or 1 * 25. Let's tackle each of these scenarios one by one.

Scenario 1: The Password is 1 * 20

If the password is 1 * 20, then the sum of the digits is 1 + * + 2 + 0 = 3 + *. Now, we need this sum to leave a remainder of 8 when divided by 9. In other words, 3 + * should be 8 more than a multiple of 9. The multiples of 9 are 9, 18, 27, and so on. So, we are looking for a number that can be expressed as 9k + 8, where k is an integer.

  • If 3 + * = 8, then * = 5. So, the password could be 1520, and * + â–² = 5 + 0 = 5.
  • If 3 + * = 17 (which is 9 + 8), then * = 14. But hold on! * has to be a single digit, so this option is a no-go.

So, in this scenario, the only possibility is that the password is 1520, and * + â–² = 5.

Scenario 2: The Password is 1 * 25

Now, let's consider the case where the password is 1 * 25. The sum of the digits is 1 + * + 2 + 5 = 8 + *. Again, this sum should leave a remainder of 8 when divided by 9. So, 8 + * should be 8 more than a multiple of 9.

  • If 8 + * = 8, then * = 0. So, the password could be 1025, and * + â–² = 0 + 5 = 5.
  • If 8 + * = 17 (which is 9 + 8), then * = 9. So, the password could be 1925, and * + â–² = 9 + 5 = 14.

In this scenario, we have two possibilities: the password could be 1025 (with * + â–² = 5) or 1925 (with * + â–² = 14).

Possible Values and Conclusion

Alright, let's gather all our findings. We've got three potential passwords that fit the bill:

  • 1520, where * + â–² = 5
  • 1025, where * + â–² = 5
  • 1925, where * + â–² = 14

So, the possible values for * + â–² are 5 and 14. Therefore, among the given options, the possible value of * + â–² is either 5 or 14. This was a super fun problem, and I hope you enjoyed cracking the code with me! Keep your eyes peeled for more puzzles, and remember, math can be a thrilling adventure if you approach it with curiosity and a bit of playfulness!

Breaking Down the Divisibility Rule of 9

The divisibility rule of 9 is a handy trick to quickly determine if a number is divisible by 9 without actually performing the division. The rule states that if the sum of the digits of a number is divisible by 9, then the number itself is also divisible by 9. Let's explore why this rule works and how we can apply it in various scenarios.

Why Does the Divisibility Rule of 9 Work?

To understand why this rule works, let's consider a number in its expanded form. Take a four-digit number, say ABCD, which can be written as:

ABCD = 1000A + 100B + 10C + D

We can rewrite each term as a multiple of 9 plus a remainder:

  • 1000A = (999 + 1)A = 999A + A
  • 100B = (99 + 1)B = 99B + B
  • 10C = (9 + 1)C = 9C + C

Now, substitute these back into the original equation:

ABCD = (999A + A) + (99B + B) + (9C + C) + D

Rearrange the terms:

ABCD = 999A + 99B + 9C + A + B + C + D

Notice that 999A, 99B, and 9C are all divisible by 9. Thus, we can rewrite the equation as:

ABCD = 9(111A + 11B + C) + (A + B + C + D)

The term 9(111A + 11B + C) is clearly divisible by 9. Therefore, the divisibility of ABCD by 9 depends only on the term (A + B + C + D), which is the sum of the digits. If (A + B + C + D) is divisible by 9, then the entire number ABCD is divisible by 9.

Applying the Rule with Remainders

In our password problem, we needed to find a number that leaves a remainder of 8 when divided by 9. This means that the sum of the digits should also leave a remainder of 8 when divided by 9. Mathematically, if N is the number and S is the sum of its digits, then:

N ≡ S (mod 9)

So, if N has a remainder of 8 when divided by 9, then S must also have a remainder of 8 when divided by 9. This is why we looked for sums that were 8 more than a multiple of 9.

Examples of the Divisibility Rule of 9

Let's look at a few examples to illustrate this rule:

  1. Number: 531

    • Sum of digits: 5 + 3 + 1 = 9
    • Since 9 is divisible by 9, 531 is divisible by 9 (531 = 9 × 59).

  2. Number: 1233

    • Sum of digits: 1 + 2 + 3 + 3 = 9
    • Since 9 is divisible by 9, 1233 is divisible by 9 (1233 = 9 × 137).

  3. Number: 2358

    • Sum of digits: 2 + 3 + 5 + 8 = 18
    • Since 18 is divisible by 9, 2358 is divisible by 9 (2358 = 9 × 262).

  4. Number: 789

    • Sum of digits: 7 + 8 + 9 = 24
    • Since 24 is not divisible by 9, 789 is not divisible by 9. When you divide 789 by 9, you get a quotient of 87 and a remainder of 6. Notice that 24 also leaves a remainder of 6 when divided by 9.

Why Use the Divisibility Rule of 9?

The divisibility rule of 9 is an efficient way to quickly check if a number is divisible by 9, especially for larger numbers. It simplifies the process and reduces the need for long division. This rule is particularly useful in problem-solving, puzzles, and mathematical games where quick calculations are necessary.

By understanding the underlying principles of the divisibility rule of 9 and practicing with examples, you can enhance your numerical skills and tackle math problems with greater confidence. Remember, math is not just about memorizing rules but about understanding the logic behind them. Keep exploring, and happy calculating!