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# If A, B and C are integers, is A + B + C even?

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If A, B and C are integers, is A + B + C even?  [#permalink]

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10 Dec 2019, 23:14
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95% (hard)

Question Stats:

42% (02:29) correct 58% (02:00) wrong based on 48 sessions

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If A, B and C are integers, is A + B + C even?

(1) A - C - B is even

(2) (A - C)/B is odd

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 02:17
2
If A, B and C are integers, is A + B + C even?
A + B + C = e. Cases are:
e + e + e = e
e + o + o = e
o + e + o = e
o + o + e = e

(1) A - C - B is even
Cases for A - B - C = e
e - e - e = e
e - o - o = e
o - e - o = e
o - o - e = e
All cases suffice the requirement.
SUFFICIENT.

(2) (A - C)/B is odd
(A - C) / B = Odd
A - C = (Odd) * B. Cases are:
e - e = (o)e
o - o = (o)e
o - e = (o)o
e - o = (o)o
Again all cases satisfy the requirement.

SUFFICIENT.

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 02:55
2
(1) A - (B+C) is even
If both A and (B+C) are odd, then both A-(B+C) and A+(B+C) are even.
If both A and (B+C) are even, then both A-(B+C) and A+(B+C) are even.
SUFFICIENT

(2) (A - C)/B is odd
If both B and (A-C) are odd, then (A-C)/B is odd and A+B+C is even.
If both B and (A-C) are even, then (A-C)/B is odd and A+B+C is even.
SUFFICIENT

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 09:48
2
If A, B and C are integers, is A + B + C even?

(Statement1) A - C - B is even
Let's say that A - C - B=2k.
A=2k+B+C --> A + B + C= 2k+B+C+ B+C= 2k+2B+2C= 2(k+B+C) -EVEN (Always YES)

Sufficient

(Statement2) $$\frac{(A - C)}{B}$$ is odd
Let's say that $$\frac{(A - C)}{B}= 2k+1$$

A= B*(2k+1)+ C

--> A + B + C =B*(2k+1)+ C+ B+C= B*(2k+1+1) + 2C= B*(2k+2)+2C= 2B*(k+1)+ 2C -EVEN (Always YES)
Sufficient

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 03:41
1
Given that A, B, and C are three positive integers.
We are to determine if A+B+C = an even number.
There two ways that this is possible. If all three numbers are even or if only one is even.

Statement 1: A - C - B is even
This is sufficient since statement 1 is only possible if one of the three integers is even or all the integers are even. And we know that A+B+C can only be even if all three numbers or if one of the integers is even. Statement 1 implies one of the three numbers is even hence sufficient.

Statement 2: (A-C)/B is odd
This is also sufficient. This because if both A and C are even, A-C is even, and to get an odd number, then B must also be even, since an even number can be an odd multiple and dividing by an appropriate even number will lead to an odd number. The converse, on the other hand, is not true. An odd number cannot be an even multiple. Secondly, When either A or C is odd, A-C will be odd, hence the only way to get (A-C)/B to be odd is for B to be odd, implying we will get one even number which is sufficient to conclude that A+B+C is even.
When both A and C are odd, then A-C is even, and as stated earlier, we can only get (A-C)/B to be odd if B is odd. So we end up with one even number and two odd numbers and A+B+B is even.

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 04:30
1
Quote:
If A, B and C are integers, is A + B + C even?

(1) A - C - B is even

(2) (A - C)/B is odd.

Statement 1:
Let A-C-B = 0, with the result that A = B+C.
If A=B+C=odd, then A + (B+C) = ODD + ODD = EVEN.
If A=B+C=even, then A + (B+C) = EVEN + EVEN = EVEN.
In each case, the answer to the question stem is YES.
SUFFICIENT.

Statement 2:
Let (A - C)/B = 1, with the result that A-C = B and A = B+C.
Same equation as that yielded by Statement 1.
Since Statement 1 is sufficient, Statement 2 must also be SUFFICIENT.

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Re: If A, B and C are integers, is A + B + C even?  [#permalink]

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11 Dec 2019, 21:05
1
a+b+c ; even ; e+e+e or o+o+e
#1
A - C - B is even
e-e-e or e-o-o
sufficient
#2
(A - C)/B is odd
e-e/e = Yes
(e-o)/o= yes
(o-o)/e = yes

we are getting values in e+e+e or o+o+e
sufficient
IMO D

If A, B and C are integers, is A + B + C even?

(1) A - C - B is even

(2) (A - C)/B is odd
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If A, B and C are integers, is A + B + C even?  [#permalink]

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Updated on: 12 Dec 2019, 02:26
Quote:
If A, B and C are integers, is A + B + C even?

(1) A - C - B is even

(2) (A - C)/B is odd

odd+-even=odd
odd+-odd=even
even+-even=even

1. o+o+o=e+o=o
2. e+e+o=e+o=o
3. e+e+e=e+e=e
4. o+o+e=e+e=e

(1) A - C - B is even sufic

a-c-b=even; case 3, and 4 fit, both give an even sum.

(2) (A - C)/B is odd sufic

e/e=[odd, even, proper fraction, undefined]
o/o=[odd, proper fraction]
e/o=[even, proper fraction]
o/e=[undefined, proper fraction]

a-c/b=odd;
b=o: a-c=odd=even-odd; a,b,c=e+o+o=even.
b=e: a-c=even=even-even or odd-odd; a,b,c=e+e+e=even, and e+o+o=even.

Ans (D)

Originally posted by exc4libur on 11 Dec 2019, 04:37.
Last edited by exc4libur on 12 Dec 2019, 02:26, edited 1 time in total.
If A, B and C are integers, is A + B + C even?   [#permalink] 11 Dec 2019, 04:37
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