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Current Student B
Joined: 29 Mar 2012
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GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 How many natural numbers that are less than 10,000 can be  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 48% (02:12) correct 52% (02:12) wrong based on 336 sessions

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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
Math Expert V
Joined: 02 Sep 2009
Posts: 59020
Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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4
8
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

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.

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  [#permalink]

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4
1
1

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.
##### General Discussion
Current Student B
Joined: 29 Mar 2012
Posts: 295
Location: India
GMAT 1: 640 Q50 V26 GMAT 2: 660 Q50 V28 GMAT 3: 730 Q50 V38 Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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Bunuel wrote:
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,
Math Expert V
Joined: 02 Sep 2009
Posts: 59020
Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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2
Bunuel wrote:
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]

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Bunuel wrote:
Bunuel wrote:
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..}

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Joined: 03 Sep 2012
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Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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3
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]

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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

there are 7 single digit natural numbers . 0 is not a natural number.
there are 7x8=56 two digit numbers
there are 7x8x8=448 three digit numbers
there are 7x8x8x8=3584 four digit numbers.
Hence, there are 4095 numbers in all.
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Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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Bunuel,

The question did not say 4 digit integers, is it right here to consider only 4 digit numbers here.

Regards,

Praveen

Bunuel 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

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.

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.
Math Expert V
Joined: 02 Sep 2009
Posts: 59020
Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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Praveengeol wrote:
Bunuel,

The question did not say 4 digit integers, is it right here to consider only 4 digit numbers here.

Regards,

Praveen

Bunuel 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

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.

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.

Positive integers less than 10,000 include single-digit integers, 2-digit integers, 3-digit integers and 4-digit integers. 8^4 gives all of them. For example, if we choose 0 for the first 3 *'s we get single-digit integers and if we choose 0 for the first 2 *'s we get 2-digit integers.

Hope it's clear.
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How many natural numbers that are less than 10,000 can be  [#permalink]

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I chose the following approach to solving this question:

Total numbers to choose from: 8
Scope: 0 - 9999

1-digit numbers: 7 (all except 0)
2-digit numbers: 7 x 8 = 56
3-digit numbers: 7 x 8 x 8 = 448
4-digit numbers: 7 x 8 x 8 x 8 = 56 x 64 = 3584

Total: 7 + 56 + 448 + 3584 = 4095

Note:
Don't make the careless mistake that I made: I first added 1 to 4095 and picked 4096 because I thought 10,000 should be included. But the question asks for less than 10,000!
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Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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vomhorizon wrote:
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) ..

You have earned this kudos, thank you for a comprehensive explanation.
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Re: How many natural numbers that are less than 10,000 can be  [#permalink]

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