Problem 36 - Basic College Mathematics

Question

Determine whether 4143 is divisible by 7

Step-by-Step Solution

Answer

No, 4143 is not divisible by 7.

Step 1: Understand the Divisibility Rule for 7

The rule states: Double the last digit of the number, subtract this from the remaining leading truncated number, and then check if the result is divisible by 7. Repeat the process if necessary.

Step 2: Apply the Rule to 4143

Double the last digit: 3 * 2 = 6. Subtract this from the remaining number: 414 - 6 = 408.

Step 3: Repeat the Divisibility Check on 408

Double the last digit of 408: 8 * 2 = 16. Subtract this from the leading number: 40 - 16 = 24.

Step 4: Check if 24 is Divisible by 7

The number 24 is not divisible by 7 since 24 divided by 7 is not an integer (24 / 7 ≈ 3.4286).

Step 5: Conclude the Result

Since 24 is not divisible by 7, the original number 4143 is also not divisible by 7.

Key Concepts

divisibility rule for 7

The divisibility rule for 7 might seem tricky at first but it's actually quite simple once you break it down. Here’s the step-by-step process:

• First, double the last digit of the number.
• Next, subtract this doubled value from the remaining part of the number.
• If the result is divisible by 7, then the original number is also divisible by 7.

This process can be repeated as needed until a smaller, more manageable number is achieved. Let’s walk through an example using the number 4143:

• Double the last digit: 3 * 2 = 6.
• Subtract from the truncated number: 414 - 6 = 408.

Now, repeat for 408:

• Double the last digit: 8 * 2 = 16.
• Subtract from the leading number: 40 - 16 = 24.

Finally, check if 24 is divisible by 7. Since it’s not, we conclude that 4143 is not divisible by 7.

basic mathematics

Understanding basic mathematics is essential for grasping more complex math topics. One fundamental skill is knowing how to work with rules for divisibility. These rules help us determine if a number is divisible by another number without actually performing full division.
In our example with 4143, we used subtraction and multiplication — operations covered in basic arithmetic. Here's a quick refresher:

• Multiplication: Doubling a number means multiplying it by 2.
• Subtraction: Removing a certain amount from a number.

Simple operations like these can quickly tell us if larger numbers are divisible by smaller ones, helping us solve problems efficiently. This foundational knowledge in basic mathematics helps build up accuracy and speed in various math applications.

number theory

Number theory is a fascinating field of mathematics that deals with the properties and relationships of numbers. Divisibility rules are a big part of number theory. They allow us to quickly determine if one number is divisible by another without performing long division.
In our example, we tested if 4143 was divisible by 7 using a specific rule. This process is a part of number theory called modular arithmetic, where numbers wrap around after reaching a certain value (in this case, multiples of 7).
Number theory also explores prime numbers, factors, and multiple other topics. Learning these concepts can enhance problem-solving skills and make math feel more like a puzzle game than a set of rigid rules. The divisibility rule for 7 is just one tool in the rich toolbox that number theory provides.