Problem 24 - Basic College Mathematics

Question

For Exercises \(17-24\), test each number for divisibility by \(2,3,4,5,6,8,9,\) and 10 . $$ 143,507 $$

Step-by-Step Solution

Answer

143,507 is not divisible by 2, 3, 4, 5, 6, 8, 9, or 10.

Step 1 - Test for Divisibility by 2

A number is divisible by 2 if its last digit is even. Check if the last digit of 143,507 is even. Since 7 is odd, 143,507 is not divisible by 2.

Step 2 - Test for Divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. Calculate the sum of digits: 1 + 4 + 3 + 5 + 0 + 7 = 20.Since 20 is not divisible by 3, 143,507 is not divisible by 3.

Step 3 - Test for Divisibility by 4

A number is divisible by 4 if the last two digits form a number that is divisible by 4. Check the last two digits, 07. Since 7 is not divisible by 4, 143,507 is not divisible by 4.

Step 4 - Test for Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5. The last digit of 143,507 is 7, so 143,507 is not divisible by 5.

Step 5 - Test for Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3. Since 143,507 is neither divisible by 2 nor 3, it is also not divisible by 6.

Step 6 - Test for Divisibility by 8

A number is divisible by 8 if the last three digits form a number that is divisible by 8. Check the last three digits, 507. Since 507 is not divisible by 8, 143,507 is not divisible by 8.

Step 7 - Test for Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. Already found the sum: 20.Since 20 is not divisible by 9, 143,507 is not divisible by 9.

Step 8 - Test for Divisibility by 10

A number is divisible by 10 if its last digit is 0. The last digit of 143,507 is 7, so 143,507 is not divisible by 10.

Key Concepts

Divisibility by 2

A number is divisible by 2 if its last digit is an even number. These even numbers are 0, 2, 4, 6, and 8. For example, the number 34 is divisible by 2 because the last digit is 4, which is even. However, a number like 143,507 is not divisible by 2 as its last digit is 7, an odd number.

Divisibility by 3

To check if a number is divisible by 3, you need to sum all its digits. If this sum is divisible by 3, then the original number is also divisible by 3. For instance, take the number 27: the digits 2 and 7 add up to 9, and since 9 is divisible by 3, 27 is too. For the number 143,507, the sum of the digits is 20 (1 + 4 + 3 + 5 + 0 + 7). Because 20 is not divisible by 3, 143,507 is also not divisible by 3.

Divisibility by 4

A number is divisible by 4 if the last two digits form a number that is divisible by 4. For instance, if a number ends in 16, 16 is divisible by 4, so the entire number is divisible by 4. Considering the number 143,507, the last two digits form the number 07, which is not divisible by 4. Therefore, 143,507 is not divisible by 4.

Divisibility by 5

A number is divisible by 5 if its last digit is 0 or 5. This rule is simple and straightforward. For example, 55 and 40 are both divisible by 5 because their last digits are 5 and 0 respectively. However, for 143,507, the last digit is 7, so it is not divisible by 5.

Divisibility by 6

A number is divisible by 6 if it is divisible by both 2 and 3. To apply this rule, you need to check both conditions. For instance, the number 54 is divisible by 6 because it is divisible by both 2 and 3. To verify 143,507: it is neither divisible by 2 (because 7 is odd) nor by 3 (sum of digits is 20), thus 143,507 is not divisible by 6.

Divisibility by 8

To determine if a number is divisible by 8, check if the last three digits form a number divisible by 8. For example, if the last three digits are 120, since 120 divided by 8 equals 15, the whole number is divisible by 8. Checking the number 143,507, the last three digits are 507, which is not divisible by 8. Hence, 143,507 is not divisible by 8.

Divisibility by 9

A number is divisible by 9 if the sum of its digits is divisible by 9. For example, the number 81: the digits 8 and 1 add up to 9, and 9 is divisible by 9, so 81 is also divisible by 9. Checking 143,507, the sum of the digits is 20, and since 20 is not divisible by 9, 143,507 is not divisible by 9.

Divisibility by 10

A number is divisible by 10 if its last digit is 0. This is a simple and clear rule. For instance, 100, 230, and 40 are all divisible by 10 because their last digits are 0. In the case of 143,507, the last digit is 7. Therefore, it isn’t divisible by 10.