Problem 35 - Basic College Mathematics

Question

To answer Exercises \(33-40\), consider the following numbers. \(\begin{array}{rrrr}305 & 313,332 & 876 & 64,000 \\ 1101 & 7624 & 1110 & 9990 \\\ 13,205 & 111,126 & 5128 & 126,111\end{array}\) Which of the above are divisible by \(6 ?\)

Step-by-Step Solution

Answer

The numbers divisible by 6 are 876, 1110, and 126,111.

Step 1 - Understanding Divisibility by 6

A number is divisible by 6 if and only if it is divisible by both 2 and 3.

Step 2 - Check Divisibility by 2

A number is divisible by 2 if its last digit is even. Identify the numbers whose last digit is even: 876, 64,000, 1101, 7624, 1110, and 126,111.

Step 3 - Check Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Calculate the sum of the digits for each number identified in Step 2.

Step 4 - Sum of Digits for Even-Ended Numbers

Calculate the sums: 876: 8 + 7 + 6 = 21 64,000: 6 + 4 = 10 7624: 7 + 6 + 2 + 4 = 19 1110: 1 + 1 + 1 + 0 = 3 126,111: 1 + 2 + 6 + 1 + 1 + 1 = 12

Step 5 - Determining Divisibility by 6

Checking the results of step 4 with divisibility by 3: 876 (sum is 21, divisible by 3) 64,000 (sum is 10, not divisible by 3) 7624 (sum is 19, not divisible by 3) 1110 (sum is 3, divisible by 3) 126,111 (sum is 12, divisible by 3)

Step 6 - Final List of Divisible Numbers

The numbers that are divisible by both 2 and 3 (hence by 6) are: 876, 1110, and 126,111.

Key Concepts

divisibility by 2

Understanding the rule for divisibility by 2 is crucial for any math student. A number is divisible by 2 if its last digit is an even number. The even numbers are 0, 2, 4, 6, and 8. This rule is straightforward and easy to check. For instance, let's look at the numbers from the exercise:

• 305 (last digit 5, not even)
• 313,332 (last digit 2, even, so divisible by 2)
• 876 (last digit 6, even, so divisible by 2)
• 64,000 (last digit 0, even, so divisible by 2)
• 1101 (last digit 1, not even)
• 7624 (last digit 4, even, so divisible by 2)
• 1110 (last digit 0, even, so divisible by 2)
• 9990 (last digit 0, even, so divisible by 2)
• 13,205 (last digit 5, not even)
• 111,126 (last digit 6, even, so divisible by 2)
• 5128 (last digit 8, even, so divisible by 2)
• 126,111 (last digit 1, not even)

You can see that some of the numbers end in even digits, which means they are divisible by 2. Identifying these numbers is the first step for combined divisibility tests.

divisibility by 3

To check if a number is divisible by 3, we need to find the sum of its digits. If the sum itself is divisible by 3, then the original number is as well. This concept might seem a bit more complex than the rule for 2, but it's also easy to apply once understood. Let's use the numbers already filtered as having even-ending digits:

• 876: Sum is 8 + 7 + 6 = 21 (divisible by 3)
• 64,000: Sum is 6 + 4 + 0 + 0 + 0 = 10 (not divisible by 3)
• 7624: Sum is 7 + 6 + 2 + 4 = 19 (not divisible by 3)
• 1110: Sum is 1 + 1 + 1 + 0 = 3 (divisible by 3)
• 126,111: Sum is 1 + 2 + 6 + 1 + 1 + 1 = 12 (divisible by 3)

By summing the individual digits and determining if the sum is divisible by 3, we find which of our even-ending numbers from the previous step also meet this criterion. This helps us narrow down the list for combined divisibility by 6.

combined divisibility

A number is divisible by 6 if it meets both the rules for 2 and 3. We already identified the numbers that are divisible by 2 (those with even last digits). Next, we checked their sums to see which of these are also divisible by 3. The filtering process can be summarized by looking at our two criteria together:

• 876: Divisible by both 2 and 3 (so by 6)
• 64,000: Divisible by 2 but not 3
• 7624: Divisible by 2 but not 3
• 1110: Divisible by both 2 and 3 (so by 6)
• 126,111: Divisible by both 2 and 3 (so by 6)

Hence, the numbers from the exercise that are divisible by 6 are 876, 1110, and 126,111. By applying the rules step by step, we simplify the task of checking divisibility, making it easier to handle even for larger numbers. Remember, the key is to break down the problem into smaller, manageable parts using these rules.