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"

Question

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"

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Analyze Consecutive Integers: Analyze the properties of three consecutive integers. Three consecutive numbers can be represented as n, n+1, and n+2, where n is an integer. Among any three consecutive integers, at least one of them must be a multiple of 3, because every third number is divisible by 3.Check Statement I: Check if statement I is true. Since one of the numbers n, n+1, or n+2 is a multiple of 3, the product P = n*(n+1)*(n+2) must also be a multiple of 3. Therefore, statement I is true.Check Statement II: Check if statement II is true. Among any three consecutive integers, there is at least one even number and one odd number. However, to ensure that the product is a multiple of 4, we need two even numbers, one of which is a multiple of 4. This is not guaranteed for every set of three consecutive numbers. For example, the consecutive numbers 1, 2, 3 do not produce a product that is a multiple of 4. Therefore, statement II is not necessarily true.Check Statement III: Check if statement III is true. Since we have already established that the product will be a multiple of 3 (statement I), we now need to check if the product will also be a multiple of 2 to confirm that it is a multiple of 6. Among any three consecutive integers, there is at least one even number, which means the product will be a multiple of 2. Therefore, the product will be a multiple of both 2 and 3, which means it is a multiple of 6. Statement III is true.

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