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How many integers between 2,000 and 3,999 have a ones digit

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How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post Updated on: 09 Jun 2018, 22:26
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How many integers between 2,000 and 3,999 have a one's digit that is a prime number?

(a) 400
(b) 412
(c) 700
(d) 800
(e) 824

Originally posted by Shrija786 on 09 Jun 2018, 21:53.
Last edited by pushpitkc on 09 Jun 2018, 22:26, edited 2 times in total.
Question formatted, OA added!
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How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 22:31
Shrija786 wrote:
How many integers between 2,000 and 3,999 have a one's digit that is a prime number?

(a) 400
(b) 412
(c) 700
(d) 800
(e) 824


In every 10 digits(0-9), there are 4 prime numbers(2,3,5, and 7) which can take the place of the one's digit.

We need to check how many such integers are there between 2000 and 3999. The first such integer is 2002
and the last such integer is 3997. Also, we will have 200 such sets between 2000 and 3999.

There are \(4*200 = 800\)(Option D) integers(between 2000 and 3999) which have a prime number in the one's digit.
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 22:39
Thank you for the answer but I have a problem with understanding the number of terms that would be coming between 2,000 and 3,999, because what is what the question has stated. We know that in order to calculate the numbers between two consecutive integers is

(last term - first term) + 1

however in this case, if it has specifically mentioned that the numbers would be between integers between 2,000 and 3,999 so won' that be 1998? I have here excluded 2000 and 3,999 to derive it.

2001 to 3998

So 3998 - 2001=1997 in which we add 1 to make it 1998.
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 22:49
Shrija786 wrote:
Thank you for the answer but I have a problem with understanding the number of terms that would be coming between 2,000 and 3,999, because what is what the question has stated. We know that in order to calculate the numbers between two consecutive integers is

(last term - first term) + 1

however in this case, if it has specifically mentioned that the numbers would be between integers between 2,000 and 3,999 so won' that be 1998? I have here excluded 2000 and 3,999 to derive it.

2001 to 3998

So 3998 - 2001=1997 in which we add 1 to make it 1998.



I agree I missed the part about the integers being between 2000 and 3999
(So, the number will not be 2000 but 1998 as you rightly pointed out)

However, the first integer between 2000 and 3999 is 2002 and the last integer in that range is 3997.
So as you rightly pointed out the number is 1998, but there are 800 integers between 2000 and 3999!

Hope this clears your confusion!
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 22:55
Apologies again, this is still not clear to me :cry: , if it is 1998, won't we then consider the 1998 and not 2000?
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How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 23:07
Shrija786 wrote:
Apologies again, this is still not clear to me :cry: , if it is 1998, won't we then consider the 1998 and not 2000?



Okay, let me try again :)

Let's list the digits between 1001 and 1009 and 3991 and 3998
2001 2002 2003 2004 2005 2006 2007 2008 2009(4 digits)

3991 3992 3993 3994 3995 3996 3997 3998(4 digit)

In the range(2010 to 3990), there are 198 such lists, each of which contains 10 integers.
4 integers in each list will have prime numbers in the one's digit(numbers ending 2,3,5,7)

So, there are 8 + 198*4 = 8 + 792 = 800 such integers. Hope this clears your confusion!
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 23:08
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We have four positions - Thousands can be filled with 2 or 3. Hundreds and Tens have 10 possibilities. Units should be a prime number - 2,3,5,7 - 4 cases.

Combinations possible - 2 * 10 * 10 * 4 = 800
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 09 Jun 2018, 23:53
Shrija786 wrote:
How many integers between 2,000 and 3,999 have a one's digit that is a prime number?

(a) 400
(b) 412
(c) 700
(d) 800
(e) 824


D.

Prime numbers: 2, 3, 5, 7

Cases:

1. _ _ _ _(2). These '_' denote the four places that we need to fill (A, B, C and D in order). Let's consider the case when the unit digit is 2, i.e. D is 2. A can be filled with only two numbers, 2 and 3 (as it is not mentioned that repetition is not allowed), as the number has to be greater than 2000 and less than 3999. The remaining two places, C and D, can be filled with any number from 0-9, making 10 cases each. Therefore, the total number of cases = 2*10*10*1 = 200 (1 because we are just considering the case when unit digit is 2.)
2. _ _ _ _(3). Repeat the above process
3. _ _ _ _(5). Repeat the above process
4. _ _ _ _(7). Repeat the above process

Total = 200+200+200+200 = 800. D.
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Re: How many integers between 2,000 and 3,999 have a ones digit  [#permalink]

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New post 10 Jun 2018, 01:50
For me, I figured out firstly that for every ten numbers (specific example is 0-9), there are four prime numbers: 2, 3, 5, and 7. As we can figure out that this is the same for any set of ten numbers ending in 0 to 9 (0-9, 10-19, 20-29, etc.) obviously 2/5th (4/10th) of the numbers satisfy the criteria. Assuming we have a total number of integers divisible by 10.

From 2000 to 3999 inclusive, we have 2000 numbers. 2/5 of those 2000 numbers would be 800 numbers, a.k.a answer D.

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Re: How many integers between 2,000 and 3,999 have a ones digit &nbs [#permalink] 10 Jun 2018, 01:50
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