5? Number: Finding Coprime Values With 12

Nov 2, 2025 by Admin 42 views

Hey guys! Today, we're diving into a fun little math problem that involves figuring out coprime numbers. Specifically, we're looking at a two-digit number in the form of '5?', where we need to find the digit that makes this number coprime with 12. Sounds interesting? Let's get started!

Understanding the Problem

So, the question we're tackling is this: We have a two-digit number '5?', and we need to figure out which digit can replace the question mark to make the entire number coprime with 12. And what does 'coprime' mean? Simply put, two numbers are coprime if their greatest common divisor (GCD) is 1. In other words, they don't share any common factors other than 1.

Now, let’s break down what we know:

The number format: We have a number that looks like 5?, where the ? is a single digit we need to find.
Coprime condition: The number '5?' must be coprime with 12.
Target: Find all possible values for '?' that satisfy the coprime condition, and then calculate the sum of these values.

Let's dive deeper into why understanding the term coprime is super important here. When we say two numbers are coprime, we mean that they don't share any common factors other than 1. For example, 8 and 15 are coprime because the factors of 8 are 1, 2, 4, and 8, while the factors of 15 are 1, 3, 5, and 15. The only common factor they share is 1. Understanding this concept is crucial because it helps us narrow down the possible values for the digit '?' in our number '5?'. We need to ensure that '5?' does not share any factors with 12 other than 1.

To solve this, we'll go through each possible digit (0-9) for the question mark and check if the resulting number is coprime with 12. It's a bit of trial and error, but we'll do it systematically to make sure we don't miss any possibilities. So, stay tuned as we explore each digit and figure out which ones make '5?' coprime with 12!

Analyzing the Coprime Condition with 12

Alright, before we start plugging in numbers, let's think about what it means to be coprime with 12. The number 12 has factors of 1, 2, 3, 4, 6, and 12. Therefore, for '5?' to be coprime with 12, it should not be divisible by 2 or 3. If a number is divisible by 2, it's even. If a number is divisible by 3, the sum of its digits is divisible by 3. Keeping these rules in mind will help us quickly eliminate possibilities.

To ensure that the number '5?' is coprime with 12, we need to make sure it does not share any factors with 12 other than 1. Since the factors of 12 are 1, 2, 3, 4, 6, and 12, the number '5?' must not be divisible by 2 or 3. If a number is divisible by 2, it is an even number. If a number is divisible by 3, the sum of its digits must be divisible by 3. Let's break this down further:

Not Divisible by 2: This means '5?' must be an odd number. So, the digit '?' can only be 1, 3, 5, 7, or 9.
Not Divisible by 3: For a number to be divisible by 3, the sum of its digits must be divisible by 3. So, 5 + ? should not be a multiple of 3. This is a crucial condition that will help us filter out more possibilities.

By keeping these two rules in mind, we can efficiently narrow down the possible values for '?' and make our task much easier. We'll proceed by checking each possible digit and seeing if it meets both criteria. Let's get to it!

Testing Possible Values for ?

Okay, let's roll up our sleeves and test each possible digit for the question mark in '5?'. We'll go through the digits 0 to 9 and see which ones make the number coprime with 12. Remember, the number should not be divisible by 2 or 3.

If ? = 0: The number is 50. It's even (divisible by 2), so it's not coprime with 12. Eliminate.
If ? = 1: The number is 51. The sum of the digits is 5 + 1 = 6, which is divisible by 3. So, 51 is divisible by 3 and not coprime with 12. Eliminate.
If ? = 2: The number is 52. It's even (divisible by 2), so it's not coprime with 12. Eliminate.
If ? = 3: The number is 53. It's odd, and the sum of the digits is 5 + 3 = 8, which is not divisible by 3. So, 53 is coprime with 12. Keep it!
If ? = 4: The number is 54. The sum of the digits is 5 + 4 = 9, which is divisible by 3. So, 54 is divisible by 3 and not coprime with 12. Eliminate.
If ? = 5: The number is 55. It's odd, and the sum of the digits is 5 + 5 = 10, which is not divisible by 3. So, 55 is coprime with 12. Keep it!
If ? = 6: The number is 56. It's even (divisible by 2), so it's not coprime with 12. Eliminate.
If ? = 7: The number is 57. The sum of the digits is 5 + 7 = 12, which is divisible by 3. So, 57 is divisible by 3 and not coprime with 12. Eliminate.
If ? = 8: The number is 58. It's even (divisible by 2), so it's not coprime with 12. Eliminate.
If ? = 9: The number is 59. It's odd, and the sum of the digits is 5 + 9 = 14, which is not divisible by 3. So, 59 is coprime with 12. Keep it!

So, after checking all the possibilities, we found that the digits 3, 5, and 9 make the number '5?' coprime with 12. Now, let's add them up!

Calculating the Sum of Possible Values

Alright, now that we've identified the possible values for the digit '?' that make the number '5?' coprime with 12, it's time to calculate their sum. We found that the digits 3, 5, and 9 satisfy the condition. So, we just need to add these numbers together.

The possible values for '?' are:

Now, let's add them up:

3 + 5 + 9 = 17

So, the sum of the possible values for '?' is 17. This means that if we replace the question mark with 3, 5, or 9, the resulting number will be coprime with 12. And that's exactly what we were trying to find!

Final Answer

Okay, guys, we've reached the end of our math adventure! We started with the problem of finding a digit to replace the question mark in '5?' to make the number coprime with 12. After breaking down the problem, understanding the coprime condition, and testing all possible values, we found that the digits 3, 5, and 9 work.

Then, we calculated the sum of these digits and found it to be 17. So, the final answer to our problem is 17.

Therefore, the sum of the possible values that '?' can take is 17. The correct answer is (d) 17.