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# What is the value of the positive two-digit integer N?

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GMATH Teacher
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What is the value of the positive two-digit integer N?  [#permalink]

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20 Feb 2019, 07:46
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85% (hard)

Question Stats:

47% (02:33) correct 53% (02:27) wrong based on 49 sessions

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GMATH practice exercise (Quant Class 16)

What is the value of the positive two-digit integer N?

(1) N and the product of the digits of N are 12 units apart.
(2) N is greater than the product of the digits of N.

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: What is the value of the positive two-digit integer N?  [#permalink]

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20 Feb 2019, 08:13
Let the number N = AB
N can be any number from 10-99.

Statement 1 -
N -A*B = 12.
N can be 28, 39...
not sufficient..

Statement 2 -
N > A*B
Not sufficient...

Combining both of them still won't help us because N can be either 28 or 39.
Therefore E is the right answer.

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Re: What is the value of the positive two-digit integer N?  [#permalink]

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20 Feb 2019, 08:13
Let the number N = 10x+y

St1)

N-xy = 12 or xy - N = 12,

Checking for N-xy = 12 ,

10x+y -xy = 12
or x(10-y)+y = 12
Put y = 0, x is not a digit
also y =1, x is not a digit
.
.
.
For y = 8, x = 2: N = 28
For y = 9, x = 39; N =39

Since we are getting two different values of N, not sufficient

St2 ) N is greater than product of digits
N can be any number such as 13, 27 etc
not sufficient

Combining St1 and St2 , we get
10x+y -xy = 12,
which gives 2 values of N as explained in St1

Hence, NOT SUFFICIENT

fskilnik wrote:
GMATH practice exercise (Quant Class 16)

What is the value of the positive two-digit integer N?

(1) N and the product of the digits of N are 12 units apart.
(2) N is greater than the product of the digits of N.

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Re: What is the value of the positive two-digit integer N?  [#permalink]

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20 Feb 2019, 10:52
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

What is the value of the positive two-digit integer N?

(1) N and the product of the digits of N are 12 units apart.
(2) N is greater than the product of the digits of N.

given
value of N ; 10a+b
#1
10a+b-ab= 12
no value of ab in sufficient
#2
10a+b>ab
again in sufficient
from 1 & 2
no clear info
in sufficient
IMO E
GMATH Teacher
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Joined: 12 Oct 2010
Posts: 935
What is the value of the positive two-digit integer N?  [#permalink]

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20 Feb 2019, 13:26
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

What is the value of the positive two-digit integer N?

(1) N and the product of the digits of N are 12 units apart.
(2) N is greater than the product of the digits of N.

$$? = N = \left\langle {ab} \right\rangle \,\,\,\,\,\,\,\left[ {a \ge 1\,,\,b \ge 0\,\,\,{\rm{digits}}} \right]$$

$$\left( 1 \right)\,\,\,\left| {\left( {10a + b} \right) - ab} \right| = 12\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,\left| {\left( {b - 10} \right)\left( {1 - a} \right)} \right| = 2\,\,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,\,\left( {10 - b} \right)\left( {a - 1} \right) = 2$$

$$\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b\,,\,\,a - 1} \right)\,\,{\rm{pair}}\,\,\,{\rm{of}}\,\,{\rm{positive}}\,\,{\rm{divisors}}\,\,{\rm{of}}\,\,2\,\,\,\,\, \Rightarrow \,\,\,\,\left( {10 - b,a - 1} \right) \in \left\{ {\left( {1,2} \right),\left( {2,1} \right)} \right\}$$

$$\left. \matrix{ \left\{ \matrix{ \,10 - b = 1 \hfill \cr \,a - 1 = 2 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {3,9} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 39\,\,{\rm{viable!}}\,\,\,\, \hfill \cr \left\{ \matrix{ \,10 - b = 2 \hfill \cr \,a - 1 = 1 \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {a,b} \right) = \left( {2,8} \right)\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{viable!}} \hfill \cr} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,N = 28\,\,{\rm{or}}\,\,39$$

$$\left( * \right)\,\,\left( {10a + b} \right) - ab = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b = 12\,\,\,\,\, \Leftrightarrow \,\,\,\,\, - a\left( {b - 10} \right) + b - \underline {10} = 12 - \underline {10} \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {b - 10} \right)\left( {1 - a} \right) = 2$$

$$\left( {**} \right)\,\,\,\left\{ \matrix{ \,b - 10 < 0\,\,\,\, \Rightarrow \,\,\,\,\left| {b - 10} \right| = 10 - b \hfill \cr \,1 - a \le 0\,\,\,\, \Rightarrow \,\,\,\,\left| {1 - a} \right| = a - 1 \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\,\left\{ \matrix{ \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 28\,\,\,\,\,\left[ {28 > 16} \right]\,\,\,\, \Rightarrow \,\,\,? = 28 \hfill \cr \,\left( {{\mathop{\rm Re}\nolimits} } \right){\rm{Take}}\,\,N = 39\,\,\,\,\,\left[ {39 > 27} \right]\,\,\,\, \Rightarrow \,\,\,? = 39 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\left( {\rm{E}} \right)$$

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

Regards,
Fabio.

P.S.: we know 28 and 39 could be found by some "organized manual work"... but the importance of this solution is contained in the techniques presented, not in the numbers themselves!
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
What is the value of the positive two-digit integer N?   [#permalink] 20 Feb 2019, 13:26
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