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If a and b are positive integers such that a < b, is b ev

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If a and b are positive integers such that a < b, is b ev

by gmattesttaker2 » Sat Sep 14, 2013 3:25 pm
Hello,

This is from MGMAT:

If a and b are positive integers such that a < b, is b even?

(1) b/2 - a/2 is an integer.

(2) 3b/4 - a/2 is an integer.


OA: B


I was trying to solve this as follows:

1) b/2 - a/2 = integer
(b - a)/2 = integer
b - a = even integer

So, b = a = odd
or b = a = even

Insuff.


2) 3b/4 - a/2 = integer
3b/4 - 2a/4 = integer
3b - 2a = even integer
3b - even integer = even integer

However, I was stuck after this point since 3b is odd.

Can you please assist? Thanks a lot - Sri
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by Brent@GMATPrepNow » Sat Sep 14, 2013 3:29 pm
gmattesttaker2 wrote: If a and b are positive integers such that a < b, is b even?

(1) b/2 - a/2 is an integer.

(2) 3b/4 - a/2 is an integer.
Target question: Is b even?

Statement 1: b/2 - a/2 is an integer
We can combine the fractions to get (b-a)/2 is an integer.
If (b-a)/2 is an integer, then b-a must be even.
So, statement 1 is really just telling us that b-a is even. There are several pairs of values that satisfy this condition. Here are two:
case a: a=3 and b=5, in which case b is not even
case b: a=2 and b=6, in which case b is even
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 3b/4 - a/2 is an integer
We can combine the fractions to get (3b-2a)/4 is an integer.
If (3b-2a)/4 is an integer, then 3b-2a must be divisible by 4.
If 3b-2a is divisible by 4, then 3b-2a must be even
Well, we know that 2a will be even for all integer values of a
So, if 3b - 2a = even, then 3b must be even.
If 3b is even, then b must be even
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer = B

Cheers,
Brent
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by Brent@GMATPrepNow » Sat Sep 14, 2013 3:33 pm
gmattesttaker2 wrote:Hello,

This is from MGMAT:

If a and b are positive integers such that a < b, is b even?

(1) b/2 - a/2 is an integer.

(2) 3b/4 - a/2 is an integer.


OA: B


I was trying to solve this as follows:

1) b/2 - a/2 = integer
(b - a)/2 = integer
b - a = even integer

So, b = a = odd
or b = a = even

Insuff.


2) 3b/4 - a/2 = integer
3b/4 - 2a/4 = integer
3b - 2a = even integer
3b - even integer = even integer

However, I was stuck after this point since 3b is odd.

Can you please assist? Thanks a lot - Sri
You were 98% there.

You got to: 3b - even integer = even integer
If something - even = even, then something is even (EVEN - EVEN = EVEN)

In other words, 3b is EVEN
If 3b is even, b must be even

Cheers,
Brent
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by gmattesttaker2 » Sat Sep 14, 2013 3:44 pm
Brent@GMATPrepNow wrote:
gmattesttaker2 wrote:Hello,

This is from MGMAT:

If a and b are positive integers such that a < b, is b even?

(1) b/2 - a/2 is an integer.

(2) 3b/4 - a/2 is an integer.


OA: B


I was trying to solve this as follows:

1) b/2 - a/2 = integer
(b - a)/2 = integer
b - a = even integer

So, b = a = odd
or b = a = even

Insuff.


2) 3b/4 - a/2 = integer
3b/4 - 2a/4 = integer
3b - 2a = even integer
3b - even integer = even integer

However, I was stuck after this point since 3b is odd.

Can you please assist? Thanks a lot - Sri
You were 98% there.

You got to: 3b - even integer = even integer
If something - even = even, then something is even (EVEN - EVEN = EVEN)

In other words, 3b is EVEN
If 3b is even, b must be even

Cheers,
Brent

Hello Brent,

Thank you very much for your reply. I was just wondering if there are any values for which 3b is even or if in the above case it is correct to use theory and conclude that b is even. Thanks for all your valuable time and help.

Best Regards,
Sri
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by Brent@GMATPrepNow » Sat Sep 14, 2013 3:54 pm
Hi Sri,

Here's what you need to know for EVEN and ODD integers.
(EVEN)(EVEN) = EVEN
(EVEN)(ODD) = EVEN
(ODD)(ODD) = ODD

Since 3 is an odd integer, and since b is an integer, b must be even.

Now, if we remove the condition that b is an integer, then b could equal a fraction such as 2/3 in which case 3b = 2.

But since we're told that b is an integer, we can be certain that b is even.

Does that help?

Cheers,
Brent
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by gmattesttaker2 » Sat Sep 14, 2013 4:15 pm
Brent@GMATPrepNow wrote:Hi Sri,

Here's what you need to know for EVEN and ODD integers.
(EVEN)(EVEN) = EVEN
(EVEN)(ODD) = EVEN
(ODD)(ODD) = ODD

Since 3 is an odd integer, and since b is an integer, b must be even.

Now, if we remove the condition that b is an integer, then b could equal a fraction such as 2/3 in which case 3b = 2.

But since we're told that b is an integer, we can be certain that b is even.

Does that help?

Cheers,
Brent

Hello Brent,

Thank you very much for your prompt reply. It is clear now. Sorry I got confused with this 3b thing. I just forgot that b has to be even for 3b to be even. Thanks for your excellent explanation (as always!) and for all your help.

Best Regards,
Sri
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