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22 Jul 2021, 01:31
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D
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Difficulty:
85%
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Question Stats:
46% (02:04) correct
54%
(02:08)
wrong
based on 127
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If a, b, and c are all integers, is ab + bc + ca + a^2 odd?
(1) a + 2b + c is even.
(2) abc is odd.
(1) a + 2b + c is even.
(2) abc is odd.
22 Jul 2021, 03:06
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Simplify :-
= ab + bc + ca + a^2
=a (a + b + c) + bc
Statement (1) a + 2b + c is even.
(a + c) must be even
i.e
Case 1: a and c both are even; b either odd or even
let us check Expression=> a (a + b + c) + bc Using E for E even and O for Odd
E( E + b + E) + b*E = E+E =E
Case 2: a and c both are odd; b either odd or even
let us check Expression=> a (a + b + c) + bc Using E for E even and O for Odd
O( O + b + O) + b*O
Say b is Odd
O( O + O + O) + O*O = O + O =E
Say b is Even
O( O + E + O) + E*O = E + E = E
I.e In any case Expression will be even
Sufficient
Statement (2) abc is odd.
i.e a, b and c are Odd
=a (a + b + c) + bc
=O(O+O+O) +O*O = O + O = E
I.e in any case expression will be even
Sufficient
(D) is the answer IMO, but surely there is a shorter way.
= ab + bc + ca + a^2
=a (a + b + c) + bc
Statement (1) a + 2b + c is even.
(a + c) must be even
i.e
Case 1: a and c both are even; b either odd or even
let us check Expression=> a (a + b + c) + bc Using E for E even and O for Odd
E( E + b + E) + b*E = E+E =E
Case 2: a and c both are odd; b either odd or even
let us check Expression=> a (a + b + c) + bc Using E for E even and O for Odd
O( O + b + O) + b*O
Say b is Odd
O( O + O + O) + O*O = O + O =E
Say b is Even
O( O + E + O) + E*O = E + E = E
I.e In any case Expression will be even
Sufficient
Statement (2) abc is odd.
i.e a, b and c are Odd
=a (a + b + c) + bc
=O(O+O+O) +O*O = O + O = E
I.e in any case expression will be even
Sufficient
(D) is the answer IMO, but surely there is a shorter way.
Bunuel
22 Jul 2021, 04:30
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Q: is ab + bc + ca + a^2 odd?
1) a + 2b + c is even => a + c is even
Rearrange: b(a+c) + a(a+c) = (a+b)(a+c) is even (sufficient)
2) abc is odd => a odd, b odd, c odd => (a+b)(a+c) even (sufficient)
Ans: D
1) a + 2b + c is even => a + c is even
Rearrange: b(a+c) + a(a+c) = (a+b)(a+c) is even (sufficient)
2) abc is odd => a odd, b odd, c odd => (a+b)(a+c) even (sufficient)
Ans: D
