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How many integers between 2,000 and 3,999 have a ones digit
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05 Aug 2018, 14:32
05 Aug 2018, 14:32
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Question Stats:
70% (01:11) correct
29% (01:22) wrong based on 47 sessions
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Re: How many integers between 2,000 and 3,999 have a ones digit
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08 Aug 2018, 18:05
08 Aug 2018, 18:05
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Expert Reply
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.
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.
Re: How many integers between 2,000 and 3,999 have a ones digit
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14 Oct 2018, 01:58
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?
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?
