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# If z is a three-digit positive integer, what is the value of

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Joined: 11 May 2008
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If z is a three-digit positive integer, what is the value of  [#permalink]

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Updated on: 31 May 2013, 04:43
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67% (01:42) correct 33% (01:30) wrong based on 520 sessions

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If z is a three-digit positive integer, what is the value of the tens digit of z ?

(1) The tens digit of z - 91 is 3
(2) The units digit of z + 9 is 5

Originally posted by arjtryarjtry on 08 Sep 2008, 23:58.
Last edited by Bunuel on 31 May 2013, 04:43, edited 2 times in total.
Edited the question, added the OA and moved to DS forum.
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Joined: 02 Sep 2009
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If z is a three-digit positive integer, what is the value of  [#permalink]

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31 May 2013, 05:04
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If z is a three-digit positive integer, what is the value of the tens digit of z ?

(1) The tens digit of z - 91 is 3 --> if z-91=130, then z=221 but if z-91=139, then z=230. Two different answers, hence this statement is not sufficient.

(2) The units digit of z + 9 is 5 --> if z+9=115, then z=106 but if z+9=125, then z=116. Two different answers, hence this statement is not sufficient.

Notice that this statement implies that the units digit of z must be 6 (in order the units digit of z+9 to be 5).

(1)+(2) From (2) we know that the units digit of z is 6. Now, (1) says that the tens digit of z-91=ab6-91 is 3 --> z=ab6=x3z+91, therefore b (the tens digit of z) must be 2. Sufficient.

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Hope it helps.
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Re: If z is a three-digit positive integer, what is the value of  [#permalink]

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03 Oct 2018, 11:54
arjtryarjtry wrote:
If z is a three-digit positive integer, what is the value of the tens digit of z ?

(1) The tens digit of z - 91 is 3
(2) The units digit of z + 9 is 5

$$\underline a \,\,\underline b \,\,\underline c \,\,\,\left\{ \matrix{ \,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\} \hfill \cr \,\,b\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr \,\,c\,\, \in \,\,\,\left\{ {\,0,1,2, \ldots ,9\,} \right\} \hfill \cr} \right.$$

$$? = b$$

$$\left( 1 \right)\,\,\, \Rightarrow \,\,\,\left\{ \matrix{ \,\underline a \,21 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\, \ldots \,\,\,;\,\,\,\,\,\underline a \,29 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,;\,\,\,\,\,\underline a \,30 - 91 = \,\,{\rm{tens}}\,\,{\rm{digit}}\,\,3 \hfill \cr {\rm{where}}\,\,a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,\,\, \hfill \cr} \right.\,\,\,$$

$$\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,1} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,3,0} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,3\,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\, \Rightarrow \,\,\,c = 6\,\,\,\,\left\{ \matrix{ \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,1,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,1\,\, \hfill \cr \,{\rm{Take}}\,\,\left( {a,b,c} \right) = \left( {1,2,6} \right)\,\,\,\, \Rightarrow \,\,\,\,\,? = \,\,2\,\, \hfill \cr} \right.$$

$$\left( {1 + 2} \right)\,\,\,c = 6\,\,\,\,\mathop \Rightarrow \limits^{\left( 1 \right)} \,\,\,\,\underline a \,26 - 91 = {\rm{tens}}\,\,{\rm{digit}}\,\,3\,\,\,\,\left( {a\,\, \in \,\,\,\left\{ {\,1,2, \ldots ,9\,} \right\}\,} \right)\,\,\,\,\, \Rightarrow \,\,\,\,b = 2\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
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Fabio Skilnik :: http://www.GMATH.net (Math for the GMAT)
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Re: If z is a three-digit positive integer, what is the value of &nbs [#permalink] 03 Oct 2018, 11:54
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