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# How many positive two-distinct-digit numbers are odd and not divisible

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Joined: 12 Oct 2010
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How many positive two-distinct-digit numbers are odd and not divisible  [#permalink]

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25 Feb 2019, 07:05
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Difficulty:

65% (hard)

Question Stats:

47% (02:10) correct 53% (02:37) wrong based on 36 sessions

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GMATH practice exercise (Quant Class 16)

How many positive two-digit numbers are odd, not divisible by 3, and have distinct digits?

(A) 28
(B) 27
(C) 26
(D) 25
(E) 24

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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
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Re: How many positive two-distinct-digit numbers are odd and not divisible  [#permalink]

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25 Feb 2019, 07:21
No of 2 digit nos from 10 to 99 = 90
45 of them are odd.
Among these odd nos multiples of 3 are 15,21,27,...99
No of odd multiples of 3 = ((99-15)/6)+1=15
So left with 45-15 = 30 digits.
Among them numbers with same digits and not multiples of 3 are 11,55 and 77...
Hence the correct answer is 30-3=27 nos...
Ans (B)......

Please corect me if I am wrong.
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Re: How many positive two-distinct-digit numbers are odd and not divisible  [#permalink]

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25 Feb 2019, 08:55
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

How many positive two-digit numbers are odd, not divisible by 3, and have distinct digits?

(A) 28
(B) 27
(C) 26
(D) 25
(E) 24

good question:
total no of 2 digit no. 90
out of which odd ; 45
we know largest 2 digit divisible by 3 ; 99, multiple 33 and 0-10 3 are multiples of 3
so 2 digit multiples of 3 = 33-3 ; 30
out which half of them would be of even two digit integer and rest odd two digit integer ; so left with 15 odd digit integers of 3
now out 45 ; 45 - 15 ; 30 are integers which are not divisible by 3
since question has asked for 2 digit integer which are distinct so 11,55,77 are removed
left with 30-3 ; 27 integers
IMO B
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Re: How many positive two-distinct-digit numbers are odd and not divisible  [#permalink]

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25 Feb 2019, 11:11
1
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

How many positive two-digit numbers are odd, not divisible by 3, and have distinct digits?

(A) 28
(B) 27
(C) 26
(D) 25
(E) 24

For any set of consecutive integers:
count = biggest - smallest + 1
Thus:
Number of two-digit integers between 10 and 99, inclusive = 99-10+1 = 90.

Of these 90 consecutive integers, 1 of every 3 will be a multiple of 3, implying that 2/3 will NOT be divisible by 3:
(2/3)(90) = 60.

Of these 60 remaining integers, exactly 1/2 will be ODD:
(1/2)(60) = 30.

Of these 30 remaining integers that are not divisible by 3 but are odd, three -- 11, 55 and 77 -- have repeated digits and thus must be subtracted from the total:
30-3 = 27.

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Re: How many positive two-distinct-digit numbers are odd and not divisible  [#permalink]

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25 Feb 2019, 13:39
fskilnik wrote:
GMATH practice exercise (Quant Class 16)

How many positive two-digit numbers are odd, not divisible by 3, and have distinct digits?

(A) 28
(B) 27
(C) 26
(D) 25
(E) 24

$$?\,\,\,\,:\,\,\,\# N\,,\,\,N \in \left[ {10,99} \right]\,\,,\,\,{\rm{odd}}\,\,{\rm{,}}\,\,{\rm{not}}\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,{\rm{3}}\,,\,\,\,{\rm{not}}\,\,{\rm{divisible}}\,\,{\rm{by}}\,\,11$$

$${\rm{I}}{\rm{.}}\,\,\,\,{\rm{odd}} \in \left[ {10,99} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{1 \over 2}\left( {99 - 10 + 1} \right) = 45\,\,{\rm{numbers}}$$

$${\rm{I}}{\rm{.}}\,\, \cap \,\,\left( {{\rm{div}}\,\,{\rm{by}}\,\,3} \right)\,\,\,:\,\,\,\left\{ \matrix{ \,15 = 3 \cdot 5 + 0 \cdot 6 \hfill \cr \,21 = 3 \cdot 5 + 1 \cdot 6 \hfill \cr \,\,\, \ldots \hfill \cr \,99 = 3 \cdot 5 + 14 \cdot 6 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,14 + 1 = 15\,\,{\rm{numbers}}$$

$${\rm{I}}{\rm{.}}\,\, \cap \,\,\left( {\underline {{\rm{not}}} \,\,{\rm{div}}\,\,{\rm{by}}\,\,3} \right)\,\,\,:\,\,\,45 - 15 = 30\,\,{\rm{numbers}}$$

$${\rm{I}}{\rm{.}}\,\, \cap \,\,\left( {\underline {{\rm{not}}} \,\,{\rm{div}}\,\,{\rm{by}}\,\,3} \right) \cap \,\,\left( {{\rm{div}}\,\,{\rm{by}}\,\,11} \right)\,\,\,:\,\,\,\left\{ {11,33,55,77,99} \right\} - \left\{ {33,99} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,3\,\,{\rm{numbers}}$$

$$?\,\, = \,\,{\rm{I}}{\rm{.}}\,\, \cap \,\,\left( {\underline {{\rm{not}}} \,\,{\rm{div}}\,\,{\rm{by}}\,\,3} \right) \cap \,\,\left( {\underline {{\rm{not}}} \,\,{\rm{div}}\,\,{\rm{by}}\,\,11} \right) = \,\,30 - 3 = 27\,\,{\rm{numbers}}$$

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

Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)
Our high-level "quant" preparation starts here: https://gmath.net
Re: How many positive two-distinct-digit numbers are odd and not divisible   [#permalink] 25 Feb 2019, 13:39
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