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How many natural numbers that are less than 10,000 can be

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How many natural numbers that are less than 10,000 can be [#permalink] New post 16 Jun 2012, 19:23
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A
B
C
D
E

Difficulty:

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Question Stats:

40% (02:50) correct 60% (02:00) wrong based on 65 sessions
How many natural numbers that are less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 & 8?

A. 5000
B. 4096
C. 6560
D. 4095
E. 8000
[Reveal] Spoiler: OA

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Re: How many natural numbers that are less than 10,000 [#permalink] New post 16 Jun 2012, 20:29
Answer D.

Solution:

We are missing digits 5 and 9 in the question. So we have 8 digits for each position (ones,tens,hundreds,thousands).

The numbers that can be constructed using 8 digits in 4 positions = (8 x 8 x 8 x 8) - 1 (To disregard the case 0000)
= 4096 - 1
= 4095

If you have use 10 digits (0 thru 9), you have (10 x 10 x 10 x 10) - 1 = 9999 numbers.
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 17 Jun 2012, 01:50
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cyberjadugar wrote:
How many natural numbers that are less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 & 8?

A. 5000
B. 4096
C. 6560
D. 4095
E. 8000


The question should read:
How many positive integers less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 and 8?

We are given 8 digits to form the numbers of a type ****. Now, each slot in **** can take 8 values, so total numbers would be 8*8*8*8=8^4, but we should exclude number 0 (since 0 is one of the digits given than 8^4 will include 0 too). So, the final answer would be 8^4-1.

Now, the question becomes how to calculate this value. The units digit of 8^4 would be 6, so the units digit of 8^4-1 would be 5. Only answer choice D fits.

Answer: D.

Similar question to practice:
how-many-four-digit-positive-integers-can-be-formed-by-using-133069.html
how-many-positive-integers-of-four-different-digits-each-100898.html
how-many-even-numbers-greater-than-300-can-be-formed-with-100578.html
how-many-even-4-digit-numbers-can-be-formed-so-that-the-95371.html
m04-70602.html

Hope it helps.
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 17 Jun 2012, 01:54
Bunuel wrote:
The question should read:
How many positive integers less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 and 8?

Hi,

I am just curious, that what difference it would make if we say positive integers or natural numbers?

Regards,
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 17 Jun 2012, 02:03
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cyberjadugar wrote:
Bunuel wrote:
The question should read:
How many positive integers less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 and 8?

Hi,

I am just curious, that what difference it would make if we say positive integers or natural numbers?

Regards,


Not much. Though notice that "there is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}" so I've never seen the official GMAT question using "natural numbers".

Hope it's clear.
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 17 Jun 2012, 02:36
Bunuel wrote:
cyberjadugar wrote:
Bunuel wrote:
The question should read:
How many positive integers less than 10,000 can be formed using the digits 0, 1, 2, 3, 4, 6, 7 and 8?

Hi,

I am just curious, that what difference it would make if we say positive integers or natural numbers?

Regards,


Not much. Though notice that "there is no universal agreement about whether to include zero in the set of natural numbers: some define the natural numbers to be the positive integers {1, 2, 3, ...}, while for others the term designates the non-negative integers {0, 1, 2, 3, ...}" so I've never seen the official GMAT question using "natural numbers".

Hope it's clear.


I always thought that
Natural numbers = {1,2,3...}
and Whole numbers = {0,1,2,3..}

Never knew about this ambiguity.
Thanks!.. :)
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 24 Sep 2012, 05:54
According to what i have read - it is always better to assume natural numbers beginning from 01 onward (for the gmat) unless otherwise stated. Assuming that to be the case we can sub divide the problem into :

A) Four digit Numbers : _ _ _ _ , The ones place - hundreds place can be filled by any one of the 8 numbers given (the problem speaks nothing about NOT REPEATING the numbers so we have to assume that they can be repeated) the thousands place can be filled by any number except "0".. This gives us 7 x 8 x 8 x 8 = 3584

B) Three digit Numbers : _ _ _ , The ones place - tens Place can be filled by any one of the 8 numbers given ( the problem speaks nothing about NOT REPEATING the numbers so we have to assume that they can be repeated) , the hundreds place can be filled by only 7 of the given 8 numbers (If we use "0" we will end up with a two digit number ). This gives us 7 x 8 x 8 = 448

C) Two digit numbers : _ _ , The ones place can be filled up by any one of the 8 numbers given , and the tens place by any 7 of the 8 ... This gives us 7 x 8 = 56

D) Assuming that zero is not a natural number , we have seven different possibilities for the one digit numbers . 7

Add A + B + C + D , This gives us 3584 + 448 + 56 + 7 = 4095 (D)

Now let us assume that ZERO is a NATURAL number (as some mathematicians do) ... A thru C is not effected by this assumption as the resultant would have meant that a four number digit with a zero at the beginning would have effectively made it a three digit no. , and a 0 to begin a three digit number would make it a two digit number etc ... The only difference including ZERO to be a natural number would have made for D , in that it would have given it 8 possibilities instead of 7 ... Which would have added 1 to our total making it 4096. Simply by looking at the answer choices we can determine that the test maker wanted natural numbers to begin from 1 and not from 0 ( as per the answer choices) ..
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Re: How many natural numbers that are less than 10,000 can be [#permalink] New post 21 Dec 2013, 08:30
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Re: How many natural numbers that are less than 10,000 can be   [#permalink] 21 Dec 2013, 08:30
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