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If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?

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If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?  [#permalink]

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New post 02 Jan 2019, 02:49
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If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?

(1) 2a + 3b is odd.

(2) (5a + 1) + (7b + 1) is even.
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Re: If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?  [#permalink]

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New post 02 Jan 2019, 03:18
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1
Solution:

Given that a & b are Positive integers. To find whether (ab)^(b+1) − a^(a+1) is Odd.

Statement 1: 2a + 3b is Odd
2a is always Even, irrespective of a being Even or Odd
Even + 3b = Odd, so b must be Odd

Case 1: a = Even, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Even - Even = Even

Case 2: a = Odd, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Odd - Odd = Even

Therefore, Statement 1 is Sufficient.

Statement 2: (5a + 1) + (7b + 1) is even, i.e 5a + 7b = Even
5a + 7b is Even if both a & b are Even or are Odd.

Case 1: a = Even, b = Even
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Even - Even = Even

Case 2: a = Odd, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Odd - Odd = Even

Therefore, Statement 2 is Sufficient.

Hence, the Correct Answer of D. Each statement is Sufficient to answer the question.

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Joined: 25 Dec 2018
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Re: If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?  [#permalink]

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New post 03 Jan 2019, 09:39
DisciplinedPrep wrote:
Solution:

Given that a & b are Positive integers. To find whether (ab)^(b+1) − a^(a+1) is Odd.

Statement 1: 2a + 3b is Odd
2a is always Even, irrespective of a being Even or Odd
Even + 3b = Odd, so b must be Odd

Case 1: a = Even, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Even - Even = Even

Case 2: a = Odd, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Odd - Odd = Even

Therefore, Statement 1 is Sufficient.

Statement 2: (5a + 1) + (7b + 1) is even, i.e 5a + 7b = Even
5a + 7b is Even if both a & b are Even or are Odd.

Case 1: a = Even, b = Even
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Even - Even = Even

Case 2: a = Odd, b = Odd
From the above condition (ab)^(b+1) − a^(a+1) is always Even, since Odd - Odd = Even

Therefore, Statement 2 is Sufficient.

Hence, the Correct Answer of D. Each statement is Sufficient to answer the question.


Thank you for the detailed explanation.
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Re: If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?  [#permalink]

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New post 02 Feb 2019, 07:30
1
akurathi12 wrote:
If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?

(1) 2a + 3b is odd.

(2) (5a + 1) + (7b + 1) is even.


from 1
2a + 3b is Odd

2a will always be even, for statement 1 to be odd, b has to be odd integers

a = 1,2,3,4,5
b = 1,3,5,7

1^2 - 1, 0 is even
2^2 - 2^3, -4 is even
6^4 - 2^3, 640 is even
the value will always be an Even value
Sufficient

(2) (5a + 1) + (7b + 1) is even.

Can be true in 2 cases
when a is odd, (5a+1) is even, when b is odd (7b+1) even => Even + even = Even
when a is even (5a+1) is odd, when b is even (7b+1) odd => Odd +odd = even

(ab)^b+1 − a^a+1
lets test some cases now
a = 1 , b = 1, 0 Even
a = 2, b = 2, Even

Again the value will always be an Even value
Sufficient

D
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Re: If a and b are positive integers, is (ab)^b+1 − a^a+1 odd?   [#permalink] 02 Feb 2019, 07:30
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