Finding Possible Values Of A Two-Digit Number's Remainder

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Hey everyone, let's dive into a cool math problem! We're going to unravel a puzzle involving remainders, multi-digit numbers, and a bit of clever thinking. The core of the question is this: We're dealing with a five-digit number, specifically 'aaa42'. When we divide this number by 48, the remainder is a two-digit number, represented as 'cd'. Our challenge? To figure out how many different values 'cd' can possibly take on. Sounds intriguing, right? Let's break it down step by step to make sure we grasp every detail.

Firstly, what exactly does 'aaa42' mean? Well, it's a five-digit number where the first three digits are identical, and the last two digits are '42'. For instance, it could be 11142, 22242, or 99942. Each of these numbers has a unique remainder when divided by 48. The concept of remainders is key here. When a number is divided by another, the remainder is what's left over after the division is performed as many times as possible without going below zero. For example, if we divide 20 by 6, we get 3 with a remainder of 2, because 6 goes into 20 three times (18), leaving 2 leftover. In our case, we are given that the remainder is a two-digit number. So our remainder, 'cd', will have a value between 10 and 99. Now we can understand the problem easily. Let's look at the possible scenarios.

To solve this, we need to focus on the divisibility rules and how remainders behave. Since 'aaa42' leaves a remainder 'cd' when divided by 48, it implies that 'aaa42' can be expressed as a multiple of 48 plus the remainder 'cd'. Mathematically, this can be represented as: aaa42 = 48 * k + cd, where 'k' is an integer (the quotient). Also, the remainder 'cd' will be a two-digit number, so 10 <= cd <= 99. Now, let's consider the possible values of the three identical digits represented by 'a'. The number 'aaa42' can be written as 10000 * a + 1000 * a + 100 * a + 42, which simplifies to 11100 * a + 42. Since we are dividing by 48, let's see what happens when we break down 11100 and 42. When we divide 11100 by 48, we get 231 with a remainder of 12. When we divide 42 by 48, we get 0 with a remainder of 42. Hence, we can write 11100 * a + 42 as (48 * 231 + 12) * a + 42, which further simplifies to 48 * 231 * a + 12 * a + 42. We can re-write this as 48 * (231 * a) + (12 * a + 42).

Unpacking the Remainder: Finding the Right Strategy

Okay, let's get to the heart of the matter: finding the possible values for 'cd'. We're dealing with the remainder when 'aaa42' is divided by 48. Knowing that 'aaa42' can be written as 11100 * a + 42, and using our previous breakdown, we can determine that the remainder will depend on the value of 'a'. Since 'a' can be any digit from 1 to 9 (it can't be 0, because then we wouldn't have a five-digit number), we'll test each value to see how it affects the remainder. It is important to understand that we have a general form for 'aaa42' which can be expressed as 48 * (231 * a) + (12 * a + 42). We know that 48 * (231 * a) is completely divisible by 48. Hence, the remainder will come from the expression 12 * a + 42. Thus, the remainder when 'aaa42' is divided by 48 will be equal to the remainder when (12 * a + 42) is divided by 48. This significantly simplifies our calculations. Now, let's consider each value of 'a' to find the corresponding 'cd'.

  • If a = 1: 12 * 1 + 42 = 54. When we divide 54 by 48, we get a remainder of 6. This gives us a remainder of 6, which is not a two-digit number. This is not a valid case. So, we will exclude this number.
  • If a = 2: 12 * 2 + 42 = 66. Dividing 66 by 48 gives us a remainder of 18. This gives us a value of 18, which is a valid two-digit number. So, cd = 18 is a possible solution.
  • If a = 3: 12 * 3 + 42 = 78. Dividing 78 by 48 gives us a remainder of 30. This gives us a value of 30, which is a valid two-digit number. So, cd = 30 is a possible solution.
  • If a = 4: 12 * 4 + 42 = 90. Dividing 90 by 48 gives us a remainder of 42. This gives us a value of 42, which is a valid two-digit number. So, cd = 42 is a possible solution.
  • If a = 5: 12 * 5 + 42 = 102. Dividing 102 by 48 gives us a remainder of 6. This result is 6, which is not a valid two-digit number.
  • If a = 6: 12 * 6 + 42 = 114. Dividing 114 by 48 gives us a remainder of 18. This gives us a value of 18, which is a valid two-digit number. So, cd = 18 is a possible solution.
  • If a = 7: 12 * 7 + 42 = 126. Dividing 126 by 48 gives us a remainder of 30. This gives us a value of 30, which is a valid two-digit number. So, cd = 30 is a possible solution.
  • If a = 8: 12 * 8 + 42 = 138. Dividing 138 by 48 gives us a remainder of 42. This gives us a value of 42, which is a valid two-digit number. So, cd = 42 is a possible solution.
  • If a = 9: 12 * 9 + 42 = 150. Dividing 150 by 48 gives us a remainder of 6. This result is 6, which is not a valid two-digit number.

Counting the Possibilities: The Final Answer

Alright, guys, we've done the heavy lifting! We've meticulously examined the different values of 'a' and found the corresponding remainders ('cd'). Now, let's count how many different values 'cd' can actually take on. Remember, 'cd' has to be a two-digit number, meaning it must be between 10 and 99. From our step-by-step analysis, we discovered the following valid values for 'cd': 18, 30, and 42. Therefore, the number of different values 'cd' can take on is 3. We found the possible values to be 18, 30, and 42. So, the correct answer is (A) 3. We did it! We successfully navigated the number maze and found the solution.

  • Step 1: Understand the Problem We started by breaking down the problem. The core is about finding remainders when a five-digit number 'aaa42' is divided by 48. The remainder is a two-digit number, 'cd'.

  • Step 2: Simplify the Number We expressed 'aaa42' as 11100 * a + 42, where 'a' is a digit from 1 to 9.

  • Step 3: Focus on the Remainder We determined the remainder depends on the remainder of (12 * a + 42) when divided by 48.

  • Step 4: Test and Find We tested each possible value of 'a' (1 to 9) to find the corresponding remainders.

  • Step 5: Count the Unique Remainders We counted the unique two-digit remainders we found (18, 30, and 42) which were 3 in total.

It was a fun journey, right? We broke down a complex problem into smaller, manageable steps, and the key was understanding remainders and how they behave with different numbers. Math can be quite an adventure when you approach it with the right mindset!

Wrapping Up: Key Takeaways and Further Exploration

So, what did we learn, friends? We've seen how to approach a number theory problem step-by-step, focusing on remainders, divisibility, and the importance of breaking down large numbers into more manageable pieces. The main trick was understanding that we were dealing with a modular arithmetic problem, where we're interested in what's