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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]  05 Aug 2018, 14:32
Expert's post
00:00

Question Stats:

83% (01:17) correct 16% (01:15) wrong based on 24 sessions
How many integers between 2,000 and 3,999 have a ones digit that is a prime number?

[Reveal] Spoiler: OA
800

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Sandy
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Retired Moderator
Joined: 07 Jun 2014
Posts: 4803
GRE 1: Q167 V156
WE: Business Development (Energy and Utilities)
Followers: 171

Kudos [?]: 2934 [0], given: 394

Re: How many integers between 2,000 and 3,999 have a ones digit [#permalink]  08 Aug 2018, 18:05
Expert's post
Explanation

This is a combinatorics problem. Make four “slots” (since the numbers are all four-digit numbers), and determine how many possibilities there are for each slot:

__ __ __ __

Since the number must begin with 2 or 3, there are two possibilities for the first slot. Because the ones digit must be prime and there are only four prime one-digit numbers (2, 3, 5, and 7), there are four possibilities for the last slot:

2........... 4
__ __ __ __

The other slots have no restrictions, so put 10 in them, since there are ten digits from 0–9:

2 10 10 4
__ __ __ __

Multiply to get 800.

Alternatively, figure out the pattern and add up the number of qualifying four-digit integers. In the first ten numbers, 2000–2009, there are exactly four numbers that have a prime units digit: 2002, 2003, 2005, and 2007.

The pattern then repeats in the next group of ten numbers, 2010–2020, and so on. In any group of ten numbers, then, four qualify. Between 2,000 and 3,999 there are $$3,999 – 2,000 + 1 = 2,000$$ numbers, or $$\frac{2000}{10}= 200$$ groups of ten numbers, so there are a total of $$400 \times 2 = 800$$ numbers that have a prime units digit.
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Sandy
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Re: How many integers between 2,000 and 3,999 have a ones digit [#permalink]  14 Oct 2018, 01:58
2. How many integers between 0 and 1570 have a prime tens digit and a prime units digit??
3. How many integers between 1 and 10^21 are such that the sum of their digits is 2?
4. How many 3-digit integers between 100 and 200 have a digit that is the average (arithmetic mean) of the other 2 digits?
5. How many integers between 324,700 and 458,600 have tens digit 2 and units digit 1?
6. How many four-digit numbers have at least one digit repeating?
7. In how many no. between 100 and 1000 at least one of their digits is 7?
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Re: How many integers between 2,000 and 3,999 have a ones digit [#permalink]  21 Jan 2019, 23:05
Expert's post
Just try this.
There are 10 potential ones digits.
there are 4 out of 10 prime numbers in that selection.
If you were able to choose all 10 numbers as a ones digit, there would be 2000 choices.
But if you were only able to choose 4 out of 10 of those numbers, then there would be 4/10*2000 =800 choices to make
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Re: How many integers between 2,000 and 3,999 have a ones digit [#permalink]  22 Jan 2019, 08:11
Expert's post
sandy wrote:
How many integers between 2,000 and 3,999 have a ones digit that is a prime number?

[Reveal] Spoiler: OA
800

With Happymathtutor on this..

There are 200 set of (0-9) unit's digits as there are 2000 integers.
Out of 10 digits, 0-9, there are 4 prime digits, 2, 3, 5 and 7, so 200 set will have 200*4=800
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Some useful Theory.
1. Arithmetic and Geometric progressions : https://greprepclub.com/forum/progressions-arithmetic-geometric-and-harmonic-11574.html#p27048
2. Effect of Arithmetic Operations on fraction : https://greprepclub.com/forum/effects-of-arithmetic-operations-on-fractions-11573.html?sid=d570445335a783891cd4d48a17db9825
3. Remainders : https://greprepclub.com/forum/remainders-what-you-should-know-11524.html
4. Number properties : https://greprepclub.com/forum/number-property-all-you-require-11518.html
5. Absolute Modulus and Inequalities : https://greprepclub.com/forum/absolute-modulus-a-better-understanding-11281.html

Re: How many integers between 2,000 and 3,999 have a ones digit   [#permalink] 22 Jan 2019, 08:11
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