Problem 1 - Basic College Mathematics

Question

For Exercises \(1-16,\) answer yes or no and give a reason based on the tests for divisibility. Determine whether 84 is divisible by \(3 .\)

Step-by-Step Solution

Answer

Yes, because the sum of the digits (12) is divisible by 3.

Step 1 - Sum the Digits of the Number

First, determine the sum of the digits of the number 84. The digits are 8 and 4. Adding them together gives: 8 + 4 = 12

Step 2 - Check Divisibility by 3

Next, check if the sum of the digits (12) is divisible by 3. A number is divisible by 3 if the sum of its digits is divisible by 3. Since: 12 ÷ 3 = 4 We see that 12 is divisible by 3.

Step 3 - Conclusion

Since the sum of the digits of 84 is 12 and 12 is divisible by 3, we conclude that 84 is also divisible by 3.

Key Concepts

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Divisibility rules are handy shortcuts that help us quickly figure out if a number can be divided by another number without actually doing the division. One commonly used rule is the divisibility rule by 3. This rule states that a number is divisible by 3 if the sum of its digits is divisible by 3. In this way, we don't have to do lengthy division calculations. Instead, we can simply check the sum of the digits.

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Understanding the sum of digits is crucial for applying many divisibility rules, including the rule for 3. Let’s break down how it works. For instance, if we take the number 84, its digits are 8 and 4. By adding these digits together, we get 8 + 4 = 12. Now, instead of dealing with the original number, we can just check whether 12 is divisible by 3. If it is, then the original number, 84, is also divisible by 3. This process simplifies checking large numbers. Remember, always sum the digits and then review the result.

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Basic mathematics often involves understanding fundamental concepts and applying simple rules systematically. Divisibility, for example, is a basic yet foundational concept in arithmetic. It paves the way for more complex topics like fractions, factors, and multiples. By mastering these fundamental rules, students can easily tackle seemingly complex problems. Always start by understanding these simple principles, like summing digits or checking divisibility, to build a strong mathematical foundation. This helps in breaking down larger, more complicated numbers into manageable parts.