Number Prop

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heshamelaziry
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Topic: Number Prop
PostSun Nov 15, 2009 2:55 am
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If the integer x is greater than 1, does x = 2?

A) x is evenly divisible by exactly two positive integers.
B) The sum of any two distinct positive factors of x is odd.

OA B

Last edited by heshamelaziry on Sun Nov 15, 2009 1:23 pm; edited 1 time in total
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PostSun Nov 15, 2009 9:20 am
heshamelaziry wrote:
If the integer x is greater than 1, does x = 2?

A) x is evenly divisible by exactly two positive integers.
B) The sum of any two distinct positive factors of x is odd.

OA C
(A) This tells us that x is prime (it's actually the definition of a prime number)
So, x could equal 2, but it could also equal 3, 5, 7 etc --> INSUFFICIENT

(B) Factor sum is odd. So, x could equal 2 (1+2=3) or x could equal 4 (1+2+4=7)--> INSUFFICIENT

(A&B): (A) tells us that x is a prime number. Now most prime numbers (except 2) are odd, which means the two factors will be odd (1 and the prime number itself), so the sum of the factors will always be even (e.g., x=7 --> factors are 1 and 7 --> sum=8)
However, the two factors of 2 (1 and 2) add to be an odd number (3). So, x must equal 2, which means(A)&(B) are sufficient.
Answer = C
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PostSun Nov 15, 2009 11:07 am
Are you sure its C? I think its B. 2 is the only even prime number that ANY factor sums will be odd...
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PostSun Nov 15, 2009 12:03 pm
okigbo wrote:
Are you sure its C? I think its B. 2 is the only even prime number that ANY factor sums will be odd...
You're absolutely right. I read it as the sum of ALL factors. I guess every word plays its role in the question Smile
The answer is, indeed, B.
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heshamelaziry
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PostSun Nov 15, 2009 1:26 pm
okigbo wrote:
Are you sure its C? I think its B. 2 is the only even prime number that ANY factor sums will be odd...
Brent wrote x could equal 4 (1+2+4=7) insuffecient. why this is wrong ?

B says any 2 positive factors " The sum of any two distinct positive factors of x is odd" this applies to 4 . B does not say prime numbers only. I don't understand
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PostSun Nov 15, 2009 6:49 pm
2+4=6 which is not odd so 4 cannot be the answer. No other number except for 2 will have ANY two factors be an odd number.
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PostSun Nov 15, 2009 9:41 pm
Good questions Hesham, keep it up.
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PostMon Nov 16, 2009 2:01 am
The sum of any two distinct +ve factors = odd

odd = even + odd or odd + even

Can we have a set of factors such that the sum of any two distinct +ve factors is odd?

Yes, it is a 2-member set that consists of "even and odd". It is {1,2}. Such a number is 2.
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