Solve 7(38 + 13): Find The Multiple!

by Luna Greco 37 views

Hey there, math enthusiasts! Today, we're diving into Problem 6, a fascinating question that challenges us to find the multiples resulting from the expression 7(38 + 13). This isn't just about crunching numbers; it's about understanding the underlying principles of arithmetic and how numbers interact. So, buckle up, and let's embark on this mathematical journey together!

Breaking Down the Problem

Before we jump into calculations, let's dissect the problem. We're given the expression 7(38 + 13), and our mission is to determine which number the result is a multiple of. The options are A) 6, B) 3, C) 4, D) 2, and E) 7. To solve this, we'll follow the order of operations (PEMDAS/BODMAS), which dictates that we first tackle the parentheses.

Step 1: Simplifying the Parentheses

First things first, let's simplify the expression inside the parentheses: 38 + 13. This is a straightforward addition, and when we add these two numbers, we get 51. So, our expression now looks like this: 7(51).

This is a crucial step, guys, because it transforms a slightly complex expression into a much simpler one. Remember, in math, breaking down problems into smaller, manageable steps is often the key to success. Now that we've simplified the parentheses, we can move on to the next operation.

Step 2: Multiplication

Now that we've simplified the parentheses, we're left with 7 multiplied by 51. Let's perform this multiplication to find the final result. When we multiply 7 by 51, we get 357. So, the value of the expression 7(38 + 13) is 357.

Multiplication is the heart of this problem, and it's essential to get this step right. If you're doing this by hand, make sure to double-check your calculations to avoid any errors. With the result in hand, we're now ready to identify the multiples.

Step 3: Identifying the Multiples

Here comes the exciting part: identifying which of the given options (6, 3, 4, 2, or 7) is a multiple of 357. To do this, we need to check if 357 is divisible by each of these numbers. A number is a multiple of another if it can be divided by that number without leaving a remainder.

  • Divisibility by 6: A number is divisible by 6 if it's divisible by both 2 and 3. 357 is not divisible by 2 because it's an odd number. So, it's not a multiple of 6.
  • 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 357 is 3 + 5 + 7 = 15, which is divisible by 3. So, 357 is a multiple of 3.
  • Divisibility by 4: A number is divisible by 4 if its last two digits are divisible by 4. The last two digits of 357 are 57, which is not divisible by 4. So, 357 is not a multiple of 4.
  • Divisibility by 2: As we already noted, 357 is an odd number, so it's not divisible by 2.
  • Divisibility by 7: To check for divisibility by 7, we can perform the division or use a divisibility rule. When we divide 357 by 7, we get 51 with no remainder. So, 357 is a multiple of 7.

Divisibility rules are your best friends here, guys. They can save you a lot of time and effort when trying to identify multiples. By applying these rules, we've narrowed down our options considerably.

The Answer

Based on our analysis, 357 is a multiple of both 3 and 7. However, the options only allow for one correct answer. Since 7 was one of the factors in the original expression (7(38 + 13)), it's the most direct and obvious multiple. Therefore, the correct answer is E) 7.

Choosing the right answer is the final step, and it's important to consider the context of the problem. In this case, the presence of 7 as a factor makes it the most logical choice.

Alternative Approach: Factoring

For those who enjoy a more algebraic approach, we can also solve this problem by factoring. We know that 357 is the result of 7 multiplied by 51. Let's break down 51 into its prime factors. 51 can be factored as 3 multiplied by 17. So, 357 can be expressed as 7 * 3 * 17.

Factoring is a powerful technique in mathematics, guys. It allows us to see the building blocks of a number and easily identify its multiples. In this case, the prime factorization of 357 clearly shows that it's a multiple of 3 and 7.

Why This Problem Matters

This problem isn't just about finding the right answer; it's about honing our mathematical skills and understanding how numbers work. It reinforces the importance of the order of operations, divisibility rules, and factoring – all essential concepts in mathematics.

Understanding these concepts will not only help you solve similar problems but also build a strong foundation for more advanced mathematical topics. Math is like a language; the more you practice, the more fluent you become.

Tips for Success

Here are a few tips to help you tackle similar problems in the future:

  1. Master the order of operations: Always follow PEMDAS/BODMAS to ensure you're performing operations in the correct order.
  2. Learn divisibility rules: Divisibility rules are shortcuts that can save you time and effort.
  3. Practice factoring: Factoring helps you understand the structure of numbers and identify their multiples.
  4. Break down complex problems: Divide complex problems into smaller, manageable steps.
  5. Double-check your work: Always take a moment to review your calculations and ensure you haven't made any errors.

Practice makes perfect, guys! The more you work on problems like this, the more confident and skilled you'll become.

Real-World Applications

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