Problem 29 - Basic College Mathematics

Question

Determine whether 26 is divisible by \(6 .\)

Step-by-Step Solution

Answer

26 is not divisible by 6.

Step 1 - Understand Divisibility Rules

To determine if 26 is divisible by 6, note that 6 is composed of the factors 2 and 3. Therefore, a number must be divisible by both 2 and 3 for it to be divisible by 6.

Step 2 - Check Divisibility by 2

A number is divisible by 2 if its last digit is 0, 2, 4, 6, or 8. The last digit of 26 is 6, so 26 is divisible by 2.

Step 3 - Check Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. For 26, the sum of its digits is 2 + 6 = 8. Since 8 is not divisible by 3, 26 is not divisible by 3.

Step 4 - Conclusion

Since 26 is not divisible by both 2 and 3, it is not divisible by 6.

Key Concepts

Divisibility by 2

To determine if a number is divisible by 2, you only need to look at its last digit. If the last digit is either 0, 2, 4, 6, or 8, then the number is divisible by 2. This rule is very simple to apply and helps quickly identify divisibility.
For example, consider the number 26. The last digit is 6. Since 6 is in the list (0, 2, 4, 6, 8), 26 is indeed divisible by 2. This means we can divide 26 by 2 without any remainder.
Another example could be the number 43. The last digit is 3, which is not in the list (0, 2, 4, 6, 8). Therefore, 43 is not divisible by 2.

Divisibility by 3

Determining if a number is divisible by 3 involves a different technique. To check if a number is divisible by 3, add all of its digits together. If the sum is divisible by 3, then the original number is also divisible by 3.
For instance, take the number 26 and add the digits: 2 + 6 = 8. Since 8 is not divisible by 3, 26 is not divisible by 3.
Let's try another example, the number 123. Adding its digits: 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is also divisible by 3. This method is quick and efficient, making it easy to determine divisibility.

Factors of Numbers

Understanding factors is crucial in determining divisibility. Factors are numbers that you can multiply together to get another number. For example, the factors of 6 are 1, 2, 3, and 6, because:
1 x 6 = 6
2 x 3 = 6
When we want to check if a number is divisible by another number, we're essentially checking if one of the number's factors matches our conditions.
To find factors efficiently, start with the smallest numbers and work your way up. Each time you find a pair of numbers that multiply to give the original number, you've found its factors.
Knowing these fundamental concepts of factors makes it significantly easier to understand and apply divisibility rules for various numbers.