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If the product of two integers is an even number and the sum of the

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If the product of two integers is an even number and the sum of the [#permalink]

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12 Oct 2017, 23:47
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If the product of two integers is an even number and the sum of the same two integers is an odd number, which of the following must be true?

A. The two integers are both odd.
B. The two integers are both even.
C. One of the two integers is odd and the other is even.
D. One of the integers is 1.
E. The two integers are consecutive.

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If the product of two integers is an even number and the sum of the [#permalink]

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Updated on: 13 Oct 2017, 07:51
1
Bunuel wrote:
If the product of two integers is an even number and the sum of the same two integers is an odd number, which of the following must be true?

A. The two integers are both odd.
B. The two integers are both even.
C. One of the two integers is odd and the other is even.
D. One of the integers is 1.
E. The two integers are consecutive.

Options for two integers whose product is even:
E / E (2 * 2 = 4)
O / E (3 * 2 = 6)
E / O (4 * 5 = 20)

Options for two integers whose sum is odd:
E / O (2 + 1 = 3)
O / E (7 + 4 = 11)

The second case limits the first. For sum to be odd, E + E is not possible. (2 + 2 = 4). When you (must) remove E + E, you are left with two integers. One is odd. The other is even.

So, one of the two integers must be even, and the other integer must be odd.

a = 3, b = 4
Even product? Yes. (3*4) = 12
Odd sum? Yes. 3+4 = 7. CORRECT

You can disprove the others.

A. The two integers are both odd.
a = 3, b = 5
Even product? No. 3*5 = 15
Odd sum? No. 3 + 5 = 8
REJECT

B. The two integers are both even.
a = 2, b = 4
Even product? Yes. 2*4 = 8
Odd sum? No. 2 + 4 = 6
REJECT

D. One of the integers is 1. The other can be odd.
From above Answer A, if both are odd, incorrect.

True, if a = 1 and b = 2, the conditions are satisfied. But if the other number is odd, conditions are not satisfied. One contrary example means the answer does not HAVE to be true. REJECT

E. The two integers are consecutive.
IF the two integers are consecutive, then they are odd and even, or vice versa, and they satisfy the prompt's conditions.

But the prompt just says "two integers." Those two integers need only be odd and even, not necessarily consecutive. They could be 1 and 200.

Edited to correct a mistaken assumption, with help from @jaissonespidey
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Originally posted by generis on 13 Oct 2017, 06:27.
Last edited by generis on 13 Oct 2017, 07:51, edited 1 time in total.
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Re: If the product of two integers is an even number and the sum of the [#permalink]

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13 Oct 2017, 07:30
2
genxer123 wrote:
Bunuel wrote:
If the product of two integers is an even number and the sum of the same two integers is an odd number, which of the following must be true?

A. The two integers are both odd.
B. The two integers are both even.
C. One of the two integers is odd and the other is even.
D. One of the integers is 1.
E. The two integers are consecutive.

Hmm. I get C AND E. I cannot figure out why they are not functionally equivalent.

Options for two integers whose product is even:
E / E (2 * 2 = 4)
O / E (3 * 2 = 6)
E / O (4 * 5 = 20)

For the sum of two integers to be odd, options are
E / O (2 + 1 = 3)
O / E (7 + 4 = 11)

The second case limits the first. For sum to be odd, E + E is not possible. (2 + 2 = 4). When you (must) remove E + E, you are left with two integers. One is odd. The other is even.

So, one of the two integers must be even, and the other integer must be odd. That is Answer C.

But by definition, consecutive integers are odd and even, or vice versa. That is Answer E

You can disprove the others.

A, B, and D are incorrect
A. The two integers are both odd.
a = 3, b = 5
Even product? No. 3*5 = 15
Odd sum? No. 3 + 5 = 8
REJECT

B. The two integers are both even.
a = 2, b = 4
Even product? Yes. 2*4 = 8
Odd sum? No. 2 + 4 = 6
REJECT

D. One of the integers is 1. The other can be odd.
From above Answer A, if both are odd, incorrect.

True, if a = 1 and b = 2, the conditions are satisfied. But if the other number is odd, conditions are not satisfied. One contrary example means the answer does not HAVE to be true. REJECT

C and E?
As far as I can tell, C and E are both true. (By definition, if integers are consecutive, one is even and one is odd. That seems to me to be the same as answer C.)

C. One of the two integers is odd and the other is even.
See analysis of E, below

E. The two integers are consecutive.
If a = 1 and b = 2
Even product? Yes. 1 * 2 = 2
Odd sum? Yes. 1 + 2 = 3

Try a = -1, b = 0
Even product? Yes. 0 * -1 = 0
Odd sum? Yes. 0 + (-1) = -1

Am I missing something?
Bunuel , are the answer choices correct?

The prompt asks which answer MUST BE TRUE. While consecutive integers will result in the stated conditions, the integers don't HAVE to be consecutive in order to obtain that result. Any odd/even combination will suffice.
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If the product of two integers is an even number and the sum of the [#permalink]

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13 Oct 2017, 07:37
2
jaissonespidey wrote:
genxer123 wrote:
Bunuel wrote:
If the product of two integers is an even number and the sum of the same two integers is an odd number, which of the following must be true?

A. The two integers are both odd.
B. The two integers are both even.
C. One of the two integers is odd and the other is even.
D. One of the integers is 1.
E. The two integers are consecutive.

Hmm. I get C AND E. I cannot figure out why they are not functionally equivalent.

Am I missing something?

The prompt asks which answer MUST BE TRUE. While consecutive integers will result in the stated conditions, the integers don't HAVE to be consecutive in order to obtain that result. Any odd/even combination will suffice.

jaissonespidey : Got it. I assumed the wrong thing, exactly backwards. That is, I assumed the truth of the answer choice. I'll edit. Thanks! And kudos.
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Re: If the product of two integers is an even number and the sum of the [#permalink]

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27 Nov 2017, 19:11
Bunuel wrote:
If the product of two integers is an even number and the sum of the same two integers is an odd number, which of the following must be true?

A. The two integers are both odd.
B. The two integers are both even.
C. One of the two integers is odd and the other is even.
D. One of the integers is 1.
E. The two integers are consecutive.

The only way for the product of two numbers to be even and the sum to be odd is if we have one even and one odd number.

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Re: If the product of two integers is an even number and the sum of the   [#permalink] 27 Nov 2017, 19:11
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