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If a and b are both positive integers, is b^(a + 1)– b*(a^b)

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If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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If a and b are both positive integers, is \(b^{a + 1}– b*(a^b)\) odd?

(1) \(a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10)\) is odd
(2) \(b^3 + 3b^2 + 5b + 7\) is odd
[Reveal] Spoiler: OA

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Last edited by chetan2u on 14 Aug 2017, 23:23, edited 2 times in total.
Edited the question.

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Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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New post 10 Jul 2011, 07:33
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AnkitK wrote:
If a and b are both positive integers, is b^(a+1)– b*(a^b)odd?
(1) a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
(2) b^3+ 3b^2+ 5b + 7 is odd


1.
a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
5a-8 odd
5*Integer-even=odd
odd*Integer=odd
Integer=odd
a=odd

b^(a+1)– b*(a^b)
Integer^(odd+1)-Integer*(Odd^Integer)
Integer^(even)-Integer*(Odd)

If Integer=Even
Even^Even-Even*(Odd)=even-even=even

If Integer=Odd
Odd^Even-Odd*(Odd)=odd-odd=even

Sufficient.

(2) b^3+ 3b^2+ 5b + 7 is odd
Integer^Odd+Odd*Integer^Even+Odd*Integer+Odd is odd

If Integer=Even
Even^Odd+Odd*Even^Even+Odd*Even+Odd
Even+Even+Even+odd=odd
b could be EVEN.

If Integer=Odd
Odd^Odd+Odd*Odd^Even+Odd*Odd+Odd
Odd+Odd+Odd+odd=EVEN
Thus, b cannot be ODD.
b=Even

b^(a+1)– b*(a^b)
Even^(Integer+Odd)-Even*Integer^Even
Even-even=even
Sufficient.

Ans: "D"
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Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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New post 10 Jul 2011, 12:55
AnkitK wrote:
If a and b are both positive integers, is b^(a+1)– b*(a^b)odd?
(1) a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
(2) b^3+ 3b^2+ 5b + 7 is odd


1. 5a-8 is odd
5a must be odd. a must be odd.
let a=1

Question: b^(a+1) - b(a^b)= b^a.b^1 - b. a^b = b(b^a-a^b) is odd?
let b=even=2
Expression: b(b^a-a^b)
2(2^1-1^2)=2=even
let b=odd=3
Expression : b(b^a-a^b)
3(3^1-1^3)=6=even.
Hence for b=even and b= odd, expression in question is even.
b(b^a-a^b) is odd? NO. Sufficient.

2. b(b^2+3b+5) + 7 is odd
even + odd = odd
hence expression b(b^2+3b+5) must be even. b must be even. let b=2
let a=even=2
Expression : b(b^a-a^b)
2(2^2-2^2)=0 = even
let a=odd=1
Expression: b(b^a-a^b)
2(2^1-1^2)=2=even
Hence for a=even and a= odd, expression in question is even.
b(b^a-a^b) is odd? NO. Sufficient.

OA D
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Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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New post 30 Jan 2017, 18:47
Cool one thank you guys!!!

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If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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New post 14 Aug 2017, 23:02
Hello Moderators,

This is a good question and worth Practicing. Do you think this question has to be made Math friendly? It took be sometime to understand what exactly are the Base and exponents here :(

Thanks In advance!

AnkitK wrote:
If a and b are both positive integers, is b^(a + 1)– b*(a^b) odd?

(1) a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd

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Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b) [#permalink]

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New post 14 Aug 2017, 23:34
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susheelh wrote:
Hello Moderators,

This is a good question and wort Practicing. Do you think this question has to be made Math friendly? It took be sometime to understand what exactly are the Base and exponents here :(

Thanks In advance!

AnkitK wrote:
If a and b are both positive integers, is b^(a + 1)– b*(a^b) odd?

(1) a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
(2) b^3 + 3b^2 + 5b + 7 is odd



Hi..
Edited...
\(b^{a+1}-b*a^b.......b(b^a-a^b)\)..
So cases..
I) if b is even, ans is NO as equation will be EVEN.
2) if b is odd, and can be yes when a is even AND no when a is odd.
3) if a is even, can be yes or no
4) if a is odd, always NO

Let's see the statements...

(1) a + (a + 4) + (a – 8 ) + (a + 6) + (a – 10) is odd
Or 5a-8 is odd...
Possible only when 5a is odd, thus a is odd.
Case 4 above.
\(b(b^a-a^b)\)
If b is odd..Odd(odd-odd)=odd*even=even
If b is even... Even (even-odd)=even*odf=even
Sufficient

(2) \(b^3 + 3b^2 + 5b + 7\) is odd
So B is even..
Case I..
Sufficient

D
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Kudos [?]: 5827 [1], given: 117

Re: If a and b are both positive integers, is b^(a + 1)– b*(a^b)   [#permalink] 14 Aug 2017, 23:34
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