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Manager  Joined: 09 Feb 2013
Posts: 111
If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 41% (02:15) correct 59% (02:30) wrong based on 178 sessions

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If a, b, and c are integers and ac + b is odd, is b odd?

(1) ac + ab is even
(2) b + c is odd

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Re: If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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If a, b, and c are integers and ac + b is odd, is b odd?

Given that $$ac + b=odd$$, determine whether b is odd.

(1) ac + ab is even --> $$ac+ab=even$$. Subtract this equation from the equation above: $$b-ab=odd-even=odd$$ --> $$b(1-a)=odd$$ --> $$b=odd$$. Sufficient.

(2) b + c is odd. Clearly insufficient.

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Re: If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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If a, b, and c are integers and ac + b is odd, is b odd?
$$ac+b=odd$$

A) ac + ab is even
$$ac+b+ac+ab=odd+even$$, $$2ac+b(1+a)=odd$$. The first term will be even so the second term must be odd in order to obtain even+odd=odd.
$$b(1+a)=odd$$, since an odd number results from odd*odd, both terms must be odd.
Sufficient

B) b + c is odd
B could be even or odd.
Not sufficient
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Re: If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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Bunuel wrote:
If a, b, and c are integers and ac + b is odd, is b odd?

Given that $$ac + b=odd$$, determine whether b is odd.

(1) ac + ab is even --> $$ac+ab=even$$. Subtract this equation from the equation above: $$b-ab=odd-even=odd$$ --> $$b(1-a)=odd$$ --> $$b=odd$$. Sufficient.

(2) b + c is odd. Clearly insufficient.

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if-x-and-y-are-integers-is-the-value-of-x-y-1-even-149236.html
if-x-y-and-z-are-integers-and-xy-z-is-an-odd-integer-is-81562.html
if-a-and-b-are-integers-is-b-even-99211.html
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is-a-b-c-even-130399.html
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Hope it helps.

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Re: If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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A) ac + ab is even
ac+b+ac+ab=odd+even, 2ac+b(1+a)=odd. The first term will be even so the second term must be odd in order to obtain even+odd=odd.
b(1+a)=odd, since an odd number results from odd*odd, both terms must be odd.
Sufficient

B) b + c is odd
B could be even or odd.
Not sufficient
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If a, b, and c are integers and ac + b is odd, is b odd?  [#permalink]

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emmak wrote:
If a, b, and c are integers and ac + b is odd, is b odd?

(1) ac + ab is even
(2) b + c is odd

Given:
a, b, and c are integers
ac + b is odd

Asked: Is b odd?

ac + b is odd
=> b is odd and ac is even
or b is even and a & c are odd

(1) ac + ab is even
a (c+b) is even => a is even or c+b is even
If a is even => b is odd
If c+b is even => c & b are even Not feasible
or c & b are odd => b is odd
SUFFICIENT

(2) b + c is odd
b is even and c is odd => a is odd
b is odd and c is even => a may or may not be even
NOT SUFFICIENT

IMO A
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