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? 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?   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"

Bytelearn · Accepted Answer

Identify Properties: Identify the properties of consecutive numbers. When we select three consecutive numbers, one of them has to be even, and one of them has to be odd. Since there are three numbers, at least one of them will also be a multiple of 3.Analyze Statement I: Analyze statement I: P is an integer multiple of 3. Since one of the three consecutive numbers must be a multiple of 3, the product P will also be a multiple of 3.Analyze Statement II: Analyze statement II: P is an integer multiple of 4. For P to be a multiple of 4, two of the three consecutive numbers must be even, which is not always the case. Therefore, P is not necessarily a multiple of 4.Analyze Statement III: Analyze statement III: P is an integer multiple of 6. Since P is a multiple of 3 (from statement I) and one of the three consecutive numbers is even, making P also a multiple of 2, P must be a multiple of 6 (because 6 = 2 * 3).

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