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Math Expert V
Joined: 02 Sep 2009
Posts: 53768
If x and y are positive odd integers, then which of the following must  [#permalink]

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15 00:00

Difficulty:   75% (hard)

Question Stats: 46% (01:17) correct 54% (01:04) wrong based on 349 sessions

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If x and y are positive odd integers, then which of the following must also be an odd integer?

I. $$x^{(y+1)}$$

II. $$x(y+1)$$

III. $$(y+1)^{(x-1)} + 1$$

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

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Math Expert V
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Posts: 53768
Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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4
Bunuel wrote:
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:
Attachment: must_be_odd.png [ 24.51 KiB | Viewed 5817 times ]

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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5
2
Answer = A. I only

$$odd^{(odd+1)} = odd^{even} = odd$$

$$odd * (odd+1) = odd * even = even$$

$$(odd+1)^{(odd-1)} + 1 = even^{even} +1 = even + 1 = odd$$

However, this contradicts for x = 3, y = 1

$$(3+1)^{1-1} + 1 = 1 + 1 = 2$$

Question asked is "must be odd", so answer = A
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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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1
Bunuel wrote:
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

I think the answer to this question is A only because anything to the power of odd will always be odd as odd*odd = odd.
In case of II y+1 will be even as odd+odd = even and from even*odd = even
In case of III anything to the power of even will churn out even only.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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1
I. x^(y+1)>> (odd)^even=odd
II. x(y+1)>>(odd)*even=even
III. (y+1)^(x-1) + 1>>(even)^even+1>>even+1>> odd if x is not equal to 1 and if x=1 then its (even)^0+1=1+!=2 even.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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1
I also say A.

I used 1 as an odd, positive number to test:
I. x^(y+1) = 1^(1+1) = 1^2=1, and a second effort: 3(1+1) = 3^2 = 9. So, yes.
II. x(y+1) = 1(1+1) = 1*2 = 2. So, no.
III. (y+1)^(x-1) + 1 = (1+1)^(1-1) +1 = 2^0 + 1 = 1+1 = 2. So, no.

ANS A
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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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Bunuel wrote:
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

The best way to do this is to assume values and put it in the equation.
Let x = 1 and y = 3
Option I: x^(y+1) = 1^(4+1) = 1^5 = 1 (ODD)
Option II: x(y+1) = 1(3 + 1) = 1(4) = 4 (EVEN)
Option III: (y+1)^(x-1) + 1 = (3+1)^(1-1) + 1 = 4^0 + 1 = 1 + 1 (EVEN)

But since we've got a power of 0, let us reconfirm with another value of x.
Let x = 3.

So, option III: (y+1)^(x-1) + 1 = (3+1)^(3-1) + 1 = 4^2 + 1 = 17 (ODD).
So option III doesn't give a unique answer.

So option I MUST always be odd.
Hence option A.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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tricky one...
we can eliminate B, C, and E right away.

1 is true, since an odd number raised to a positive number power will always be odd.
the third one, if x>1, then it will be odd, but if x=1, then everything is equal to 2, and not odd.

thus, 1 alone is correct.
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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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Bunuel wrote:
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

I tested the three statements choosing positive, odd values for x and y. I used x=1 and y=3 first, then x=3 y=5 next. I realized about halfway through that number properties could've done it even quicker!
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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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Option A

x,y are positive Odd integers. Odd: O, Even: E & Fraction: F

I. x^(y+1) = O^(O + 1) = O^E = O
II. x(y+1) = O(O + 1) = O(E) = E
III. (y+1)^(x-1) + 1 = (O + 1)^(O - 1) + 1 = (E)^(E) + 1 = E + 1 = O || Spl. Case, if x = 1 then (y+1)^(x-1) + 1 = (y+1)^(0) + 1 = 1 + 1 = 2 =E
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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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1
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

i) always odd as odd raise to anything which is even or odd= odd
ii) x(y+1) , since y is odd so y+1 will be even and => even x odd = even
iii) (y+1)^(x11) + 1
let y =1 , x= 1 both odd => 2^0 +1 = 2 even

Hence i) is always odd.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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Bunuel wrote:
If x and y are positive odd integers, then which of the following must also be an odd integer?

I. x^(y+1)
II. x(y+1)
III. (y+1)^(x-1) + 1

A. I only
B. II only
C. III only
D. I and III
E. None of the above

Kudos for a correct solution.

x is odd, so (x-1) is even.
y is odd, so (y+1) is even.

I.$$x^(y+1) = {odd} ^ {even} = odd$$
II. $$x(y+1) = odd * even = even$$
III. $$(y+1)^(x-1) + 1 = {even}^{even} + 1$$ OR $${even}^0 +1$$ = odd OR even

So, Only I is correct.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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No calculations required.

I. $$x^{(y+1)}$$=$$odd^{(odd+1)}$$-any odd no raised to any odd or even power will always be odd.

II. x(y+1)=odd(odd+1)=odd(even)-any number multiplyed with even no will always be even

III. $$(y+1)^{x-1}$$ + 1=

Let x=3(odd)

$$(odd+1)^{(odd-1)}$$+1=$$(even)^{(3-1)}$$+1=odd

or

Let x=1(odd)

$$even^{(1-1)}$$+1=$$even^{(0)}$$+1=1+1=2(any integer raised to zero is 1) even

Hence 3rd option can be odd or even.

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Re: If x and y are positive odd integers, then which of the following must  [#permalink]

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# If x and y are positive odd integers, then which of the following must

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