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Question 253000: Which of the following must be true?
I. The sum of two consecutive integers is odd
II. The sum of three consecutive integers is even
III.The sum of three consecutive integers is a multiple of 3
(A) I only (B) II only (C) I and II only (D) I and III only (E) I, II, and III
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! I would say selection D is the right answer.
That would be statements I and III only.
first statement says sum of 2 consecutive integers is odd.
x = integer
x + 1 = next consecutive integers.
x + (x + 1) = 2x + 1
2x is always even.
2x + 1 must always be odd.
example:
let x = 3
2x + 1 = 6 + 1 = 7 (odd)
let x = 4
2x + 1= 8 + 1 = 9 (odd)
wheether x is odd or even, 2x+1 is always odd.
first statement is true.
second statement says sum of 3 consecutive integers is even.
x + (x + 1) + (x + 2) = 3x + 3
3x can be odd or even.
3x + 3 can therefore be odd or even.
example:
let x = 3,
3x+3 = 9 + 3 = 12 (even)
let x = 4
3x+3 = 12 + 3 = 15 (odd)
second statement is false.
third statement says sum of three consecutive integers is a multiple of 3.
if this is true, then the sum of of 3 consecutive integers = 3 * y where y is an integer.
sum of 3 consecutive integers is equal to 3x+3.
(3x + 3) / 3 = (x + 1)
multiply both sides of this equation by 3 to get:
(x + 1) * 3 = 3x+3 which means that 3x+3 is a product of 3 * (x+1). This makes it a multiple of 3.
example:
let x = 3
3x+3 = 12
12/3 = 4 which is the same as x+1.
let x = 4
3x+3 = 15
15/3 = 5 which is the same as x + 1.
just to be sure, take a negative number for x.
let x = -35
3x+3 = -105 + 3 = -102
-102/3 = -34 which is the same as x + 1.
third statement is true.
I believe selection D is confirmed.
your answer is selection D.
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