megafan wrote:
If the sum of 5 consecutive integers is x, which of the following must be true?
I. x is an even number
II. x is an odd number
III. x is a multiple of 5
(A) I only
(B) II only
(C) III only
(D) I and III
(E) II and III
Let n = the smallest of the five integers
So, n + 1 = the next integer
So, n + 2 = the next integer
So, n + 3 = the next integer
So, n + 4 = the last integer
So the sum of all five integers = n + (n + 1) + (n + 2) + (n + 3) + (n + 4) =
5n + 20 In other words,
x = 5n + 20 Now let's analyze each statement...
I. x is an even number
If n = 1, then
x = 5n + 20 = 5(1) + 20 = 25, and 25 is not even.
So, statement I is not necessarily true, which means we can eliminate answer choices A and D
II. x is an odd number
If n = 2, then
x = 5n + 20 = 5(2) + 20 = 30, and 30 is not odd.
So, statement II is not necessarily true, which means we can eliminate answer choices B and E
Important: As you can see, we've already determined that the correct answer is C, without even analyzing statement III.
So, when answering Roman numeral questions such as this, be sure to eliminate answer choices along the way. Let's analyze statement III for "fun".
III. x is a multiple of 5
We already determined that
x = 5n + 20 We can factor the right hand side as follows:
x = 5(n + 4)So, we can see that x is definitely divisible by
5, which means statement III is true.
Answer: C
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