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# If G is a function defined in the positive integers, such that G(k) is

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GMATH Teacher
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Joined: 12 Oct 2010
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If G is a function defined in the positive integers, such that G(k) is  [#permalink]

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24 Feb 2019, 14:17
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75% (hard)

Question Stats:

45% (02:04) correct 55% (01:20) wrong based on 29 sessions

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GMATH practice exercise (Quant Class 14)

If $$G$$ is a function defined in the positive integers, such that $$G(k)$$ is a positive integer for each positive integer $$k$$, what is the value of $$G(G(2019))$$ ?

(1) $$G(G(m+n)) = m+n$$ , for every positive integers $$m,n$$.
(2) $$G(n) = m$$ implies $$G(m) = n$$, for every positive integers $$m,n$$.

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
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Re: If G is a function defined in the positive integers, such that G(k) is  [#permalink]

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24 Feb 2019, 14:46
Ok so let’s consider statement 1 alone. G(m+n) = m+n. So i can say that G(2019)=2019. This is beacuse lets say i express 2019 as the sum of 2 positive integers x and y. So 2019=x+y. G(2019)=G(x+y)=x+y=2019. Thus G(G(2019))=G(2019)=2019. Thus this statement is enough.

Now on to statement 2 alone. G(2019)=k (say). So G(k)=2019 ( according to this statement).Thus G(G(2019))=G(k)=2019. Thus this statement is also enough.

Cheers

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Re: If G is a function defined in the positive integers, such that G(k) is  [#permalink]

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25 Feb 2019, 12:46
fskilnik wrote:
GMATH practice exercise (Quant Class 14)

If $$G$$ is a function defined in the positive integers, such that $$G(k)$$ is a positive integer for each positive integer $$k$$, what is the value of $$G(G(2019))$$ ?

(1) $$G(G(m+n)) = m+n$$ , for every positive integers $$m,n$$.
(2) $$G(n) = m$$ implies $$G(m) = n$$, for every positive integers $$m,n$$.

$$? = G\left( {G\left( {2019} \right)} \right)$$

$$\left( 1 \right)\,\,\,\left\{ \matrix{ \,G\left( {G\left( {m + n} \right)} \right) = m + n\,\,\,\left( {m,n \ge 1\,\,{\rm{ints}}} \right) \hfill \cr \,\,\,\,\,\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2018,1} \right) \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,?\,\, = \,\,G\left( {G\left( {2018 + 1} \right)} \right)\,\, = \,\,2018 + 1\,\, = \,\,2019$$

$$\left( 2 \right)\,\,\,G\left( n \right) = m\,\,\,\, \Rightarrow \,\,\,\,\,G\left( m \right) = n\,\,\,\,\left( {m,n \ge 1\,\,{\rm{ints}}} \right)\,\,\,\,\,\left( * \right)\,$$

$${\rm{Take}}\,\,\left( {n,m} \right) = \left( {2019,G\left( {2019} \right)} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,? = G\left( {G\left( {2019} \right)} \right) = 2019$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.
_________________
Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: If G is a function defined in the positive integers, such that G(k) is   [#permalink] 25 Feb 2019, 12:46
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