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If G is a function defined in the positive integers, such th

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by fskilnik@GMATH » Sun Feb 24, 2019 1:11 pm

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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.

Answer: [spoiler]____(D)__[/spoiler]
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by fskilnik@GMATH » Mon Feb 25, 2019 11:35 am
fskilnik@GMATH 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.
$$? = G\left( {G\left( {2019} \right)} \right)$$

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

$$\left( 2 \right)\,\,\,G\left( n \right) = m\,\,\,\, \Rightarrow \,\,\,\,\,G\left( m \right) = n\,\,\,\,\left( {m,n \ge 1\,\,{\rm{ints}}} \right)\,\,\,\,\,\left( * \right)\,$$
$${\rm{Take}}\,\,\left( {n,m} \right) = \left( {2019,G\left( {2019} \right)} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,? = G\left( {G\left( {2019} \right)} \right) = 2019$$


The correct answer is (D).


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

Regards,
Fabio.
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