"15. Three consecutive numbers are selected from the set of integers from 1 to 30. Suppose P is the product of the numbers drawn. Which of the following must be true? [Without calculator] I. P is an integer multiple of 3. II. P is an integer multiple of 4. III. P is an integer multiple of 6. (A) Only I (B) Only II (C) Both I and III (D) Both II and III"

Divisibility by 3: Analyze the divisibility by 3 for three consecutive integers. In any set of three consecutive integers, at least one of them must be divisible by 3, because every third number is a multiple of 3.Divisibility by 4: Analyze the divisibility by 4 for three consecutive integers. In any set of three consecutive integers, it is not guaranteed that one of them will be a multiple of 4. For example, the consecutive numbers 1, 2, and 3 are not divisible by 4.Divisibility by 6: Analyze the divisibility by 6 for three consecutive integers. Since we have already established that at least one of the three consecutive numbers is divisible by 3, we need to check if one of the remaining two numbers is even, which would make the product divisible by 2 as well. In any set of three consecutive integers, there will always be at least one even number, thus making the product divisible by 6.Combine Results: Combine the results from Steps 1, 2, and 3 to determine which statements are true. From Step 1, we know that statement I is true. From Step 2, we know that statement II is not necessarily true. From Step 3, we know that statement III is true. Therefore, the correct answer is that both statements I and III must be true.

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Three consecutive numbers are selected from the set of integers from $1$ to $30$ . Suppose $P$ is the product of the numbers drawn. Which of the following must be true? [Without calculator] $\newline$ I. $P$ is an integer multiple of $3$ . $\newline$ II. $P$ is an integer multiple of $4$ . $\newline$ III. $P$ is an integer multiple of $6$ . $\newline$ (A) Only I $\newline$ (B) Only II $\newline$ (C) Both I and III $\newline$ (D) Both II and III

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Q. Three consecutive numbers are selected from the set of integers from

1

30

. Suppose

P

is the product of the numbers drawn. Which of the following must be true? [Without calculator]

\newline

P

is an integer multiple of

3

\newline

II.

P

is an integer multiple of

4

\newline

III.

P

is an integer multiple of

6

\newline

(A) Only I

\newline

(B) Only II

\newline

\newline

(D) Both II and III

Divisibility by $3$ : Analyze the divisibility by $3$ for three consecutive integers. In any set of three consecutive integers, at least one of them must be divisible by $3$ , because every third number is a multiple of $3$ .
Divisibility by $4$ : Analyze the divisibility by $4$ for three consecutive integers. $\newline$ In any set of three consecutive integers, it is not guaranteed that one of them will be a multiple of $4$ . For example, the consecutive numbers $1$ , $2$ , and $3$ are not divisible by $4$ .
Divisibility by $6$ : Analyze the divisibility by $6$ for three consecutive integers. Since we have already established that at least one of the three consecutive numbers is divisible by $3$ , we need to check if one of the remaining two numbers is even, which would make the product divisible by $2$ as well. In any set of three consecutive integers, there will always be at least one even number, thus making the product divisible by $6$ .
Combine Results: Combine the results from Steps $1$ , $2$ , and $3$ to determine which statements are true. $\newline$ From Step $1$ , we know that statement I is true. $\newline$ From Step $2$ , we know that statement II is not necessarily true. $\newline$ From Step $3$ , we know that statement III is true. $\newline$ Therefore, the correct answer is that both statements I and III must be true.