Questions: Suppose we know that the *sum* of the digits of a number is 24. Use appropriate divisibility tests to select all the statements that we know *must* be true. divisible by 9 divisible by 4 divisible by 2 divisible by 3 divisible by 5 divisible by 10

Solution Steps

To determine which divisibility rules apply based on the sum of the digits being 24, we can use the following rules:

1. A number is divisible by 9 if the sum of its digits is divisible by 9.
2. A number is divisible by 3 if the sum of its digits is divisible by 3.
3. Divisibility by 2, 4, 5, and 10 depends on the last digit(s) of the number, which we do not have information about.

Given the sum of the digits is 24, we can check for divisibility by 9 and 3.

To determine if the number is divisible by 9, we check if the sum of the digits \( S \) is divisible by 9. Given \( S = 24 \):

\[ 24 \mod 9 = 6 \quad (\text{not divisible}) \]

Thus, the number is not divisible by 9.

Next, we check if the number is divisible by 3. We perform the same modulus operation:

\[ 24 \mod 3 = 0 \quad (\text{divisible}) \]

Thus, the number is divisible by 3.

Since we do not have information about the last digit(s) of the number, we cannot determine divisibility by 2, 4, 5, or 10. Therefore, we conclude that:

• The number is not divisible by 9.
• The number is divisible by 3.

Final Answer