Problem Solving

BTGModeratorVI wrote: ↑

Sat Feb 15, 2020 3:30 pm

If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

I. x + y is even
II. (x+z)/y is an integer
III. xz is even

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Source: Official Guide
Answer: D

Say x = 2n + 1, where n = positive integer

Assuming x = 2n + 1 will ensure that for any positive integer value of n, x is odd.

Thus, y = x + 2 = 2n + 1 + 2 = 2n + 3 and z = y + 2 = 2n + 3 + 2 = 2n + 5

Let's take each statement one by one.

I. x + y is even: x + y = (2n + 1) + (2n + 3) = 4n + 4, an even number. True.

II. (x+z)/y is an integer: (x + z)/y = [(2n + 1) + (2n + 5)]/(2n + 3) = (4n + 6)/(2n + 3) = 2, an integer. True

III. xz is even: We know that product two odd integers is odd. Thus, this statement must be false.

The correct answer: D

Hope this helps!

-Jay
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BTGModeratorVI wrote: ↑

Sat Feb 15, 2020 3:30 pm

If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

I. x + y is even
II. (x+z)/y is an integer
III. xz is even

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Source: Official Guide
Answer: D

Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN


The key word here is MUST

I. x + y is even
Since x and y are both ODD, we can conclude that x + y = ODD + ODD = EVEN
So, statement I is true
Check the answer choices.... and ELIMINATE B and C, since they state that statement I is not true.

II. (x + z)/y is an integer
Must this be true?
Since x, y and z are consecutive ODD integers, we know that y is 2 greater than x, and z is 4 greater than x.
So, we can write the following:
x = x
y = x + 2
z = x + 4

This means that (x + z)/y = (x + x + 4)/(x + 2)
= (2x + 4)/(x + 2)
= 2
Aha, so (x + z)/y will ALWAYS equal 2 (an integer)
So, statement II is true
Check the answer choices.... and ELIMINATE A, since it states that statement II is not true.

III. xz is even
Since x and z are both ODD, we know that xz = (ODD)(ODD) = ODD
So, statement III is NOT true

Answer: D

Cheers,
Brent

BTGModeratorVI wrote: ↑

Sat Feb 15, 2020 3:30 pm

If x, y, and z are consecutive odd integers, with x < y < z, then which of the following must be true?

I. x + y is even
II. (x+z)/y is an integer
III. xz is even

A. I only
B. II only
C. III only
D. I and II only
E. I, II, and III

Source: Official Guide
Answer: D

Since x, y, and z are consecutive odd integers, x + y = odd + odd = even, and xz = (odd)(odd) = odd. So statement I is true, and statement III is false.

Since x = y - 2 and z = y + 2, then (x + z)/y = (y - 2 + y + 2)/y = 2y/y = 2 is an integer. So statement II is true also.

Answer: D