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The function f is defined for all positive integers n > 4 as [#permalink]
15 Jan 2014, 11:00

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Difficulty:

65% (hard)

Question Stats:

43% (03:10) correct
57% (02:02) wrong based on 98 sessions

The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

Re: The function f is defined for all positive integers n > 4 as [#permalink]
15 Jan 2014, 17:09

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Expert's post

Rock750 wrote:

The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

(1) f(f(a)) = a

(2) f(f(f(a))) is odd.

I'm happy to help. As usual, this is a spectacularly clever problem from those folks at MGMAT.

Notice that for this particular function, when f(n) has an even input, it yields an odd output, and vice versa: when it has an odd input, it yield an even output.

Statement #1: Well, if a is even, then f(a) = 2a - 7, which will be odd, and f(f(a)) = 3(2a - 7) - 9 = 6a - 30. Then, if f(f(a)) = a, we have 6a - 30 = a 5a = 30 a = 6 That's one possible value. If a is odd, then f(a) = 3a - 9, which will be even, and f(f(a)) = 2(3a - 9) - 7 = 6a - 25 Then, if f(f(a)) = a, we have 6a - 25 = a 5a = 25 a = 5 That's also one possible value. This statement yields two possible values, so no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Statement #2: If f(f(f(a))) is odd, then f(f(a)) is even, and f(a) is odd, and a is even. This tells us that a is even, but a could be any even number. This statement yields no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Combined We have two values from statement #1. From statement #2, we know a must be even. This means that a = 6. Now, we can give a definitive answer to the prompt question. Combined, the statements are sufficient. Answer = (C)

Re: The function f is defined for all positive integers n > 4 as [#permalink]
11 May 2014, 13:59

mikemcgarry wrote:

Rock750 wrote:

The function f is defined for all positive integers n > 4 as f(n) = 3n – 9 if n is odd and f(n) = 2n – 7 if n is even. What is the value of the positive integer a?

(1) f(f(a)) = a

(2) f(f(f(a))) is odd.

I'm happy to help. As usual, this is a spectacularly clever problem from those folks at MGMAT.

Notice that for this particular function, when f(n) has an even input, it yields an odd output, and vice versa: when it has an odd input, it yield an even output.

Statement #1: Well, if a is even, then f(a) = 2a - 7, which will be odd, and f(f(a)) = 3(2a - 7) - 9 = 6a - 30. Then, if f(f(a)) = a, we have 6a - 30 = a 5a = 30 a = 6 That's one possible value. If a is odd, then f(a) = 3a - 9, which will be even, and f(f(a)) = 2(3a - 9) - 7 = 6a - 25 Then, if f(f(a)) = a, we have 6a - 25 = a 5a = 25 a = 5 That's also one possible value. This statement yields two possible values, so no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Statement #2: If f(f(f(a))) is odd, then f(f(a)) is even, and f(a) is odd, and a is even. This tells us that a is even, but a could be any even number. This statement yields no definitive answer to the prompt. This statement, alone and by itself, is insufficient.

Combined We have two values from statement #1. From statement #2, we know a must be even. This means that a = 6. Now, we can give a definitive answer to the prompt question. Combined, the statements are sufficient. Answer = (C)

Does all this make sense? Mike

Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5,6 and 7,8 and my values are all over?

Re: The function f is defined for all positive integers n > 4 as [#permalink]
11 May 2014, 21:27

Expert's post

russ9 wrote:

Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5, 6 and 7, 8 and my values are all over?

Thanks

Dear russ9 Think about the prompt question, "What is the value of a?" It may be that a has just one value, or more than one. If you find one value, that's absolutely no guarantee that there aren't other values that also work. Suppose, for the sake of argument, that the two values that worked were a = 6 and a = 50 --- plugging in numbers for some single digit cases would never tell you that there's more than one answer. Do you see what I mean?

Remember, GMAT DS is NOT about "find the answer" --- it's more about "is it possible to find a unique and sensible answer?" If you were looking for one and only one answer, then plugging in numbers would make sense --- that might not be so bad on GMAT PS. But on GMAT DS, that misses the point in a problem such as this.

Re: The function f is defined for all positive integers n > 4 as [#permalink]
15 May 2014, 16:22

mikemcgarry wrote:

russ9 wrote:

Hi Mike,

is there a reason why choosing numbers here doesn't work or isn't optimal?

I tried use 5, 6 and 7, 8 and my values are all over?

Thanks

Dear russ9 Think about the prompt question, "What is the value of a?" It may be that a has just one value, or more than one. If you find one value, that's absolutely no guarantee that there aren't other values that also work. Suppose, for the sake of argument, that the two values that worked were a = 6 and a = 50 --- plugging in numbers for some single digit cases would never tell you that there's more than one answer. Do you see what I mean?

Remember, GMAT DS is NOT about "find the answer" --- it's more about "is it possible to find a unique and sensible answer?" If you were looking for one and only one answer, then plugging in numbers would make sense --- that might not be so bad on GMAT PS. But on GMAT DS, that misses the point in a problem such as this.

Does this make sense? Mike

Makes total sense. Thanks, Mike!

gmatclubot

Re: The function f is defined for all positive integers n > 4 as
[#permalink]
15 May 2014, 16:22