Problem 21 - Basic College Mathematics

Question

For Exercises \(17-24\), test each number for divisibility by \(2,3,4,5,6,8,9,\) and 10 . $$ 127,575 $$

Step-by-Step Solution

Answer

127,575 is divisible by 3, 5, and 9.

Step 1 - Divisibility by 2

A number is divisible by 2 if it is even (i.e., its last digit is 0, 2, 4, 6, or 8). The number 127,575 ends in 5, so it is not divisible by 2.

Step 2 - Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Calculate the sum: 1 + 2 + 7 + 5 + 7 + 5 = 27. Since 27 is divisible by 3, 127,575 is also divisible by 3.

Step 3 - Divisibility by 4

A number is divisible by 4 if the last two digits form a number that is divisible by 4. The last two digits of 127,575 are 75, which is not divisible by 4. Therefore, 127,575 is not divisible by 4.

Step 4 - Divisibility by 5

A number is divisible by 5 if it ends in 0 or 5. Since 127,575 ends in 5, it is divisible by 5.

Step 5 - Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3. 127,575 is not divisible by 2 but is divisible by 3. Therefore, it is not divisible by 6.

Step 6 - Divisibility by 8

A number is divisible by 8 if the last three digits form a number that is divisible by 8. The last three digits of 127,575 are 575. Check to see if 575 is divisible by 8. 575 / 8 = 71.875, which is not an integer, so 127,575 is not divisible by 8.

Step 7 - Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. The sum of the digits is 27, which is divisible by 9. Therefore, 127,575 is divisible by 9.

Step 8 - Divisibility by 10

A number is divisible by 10 if it ends in 0. Since 127,575 does not end in 0, it is not divisible by 10.

Key Concepts

divisibility by 2

To determine if a number is divisible by 2, check if it ends in an even digit: 0, 2, 4, 6, or 8. For example, 127,575 ends in 5, which is an odd number. Therefore, 127,575 is not divisible by 2. This rule applies to any number; just look at the last digit. If it's even, the number passes the test for divisibility by 2.

divisibility by 3

For a number to be divisible by 3, the sum of its digits must also be divisible by 3. Let's take 127,575 as an example. The sum of its digits is calculated as follows: 1 + 2 + 7 + 5 + 7 + 5 = 27. Since 27 is divisible by 3, we can conclude that 127,575 is also divisible by 3. Always remember to add all the digits together and check if the result can be divided by 3.

divisibility by 5

A number is divisible by 5 if it ends in either 0 or 5. For example, take the number 127,575, which ends in 5. Therefore, we can easily see that it is divisible by 5. This rule is simple and quick to apply, making it easy to check for divisibility by 5 in any given number.

divisibility by 9

The rule for divisibility by 9 is similar to that for 3. A number is divisible by 9 if the sum of its digits is divisible by 9. Using our example, 127,575: Calculate the sum of its digits: 1 + 2 + 7 + 5 + 7 + 5 = 27. Since 27 is divisible by 9, this means that 127,575 is also divisible by 9. This method provides a straightforward way to test large numbers for divisibility by 9.

elementary number theory

Elementary number theory covers various fundamental concepts, one of the most useful ones being divisibility rules. These rules allow us to determine if one number can be evenly divided by another without performing the actual division. Understanding divisibility helps in simplifying complex mathematical problems and plays a key role in higher areas of mathematics like factorization. Basic rules, such as checking the last digit or summing digits, make it easier to determine the properties of numbers quickly and efficiently without extensive calculations.