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How many integers between 0 and 1570 have a prime tens digit
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Updated on: 15 Oct 2013, 13:48
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How many integers between 0 and 1570 have a prime tens digit and a prime units digit? (A) 295 (B) 252 (C) 236 (D) 96 (E) 76
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Originally posted by hardworker_indian on 04 Oct 2004, 17:52.
Last edited by Bunuel on 15 Oct 2013, 13:48, edited 1 time in total.
Added the OA.




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Re: How many integers between 0 and 1570 have a prime tens digit
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15 Oct 2013, 13:47
hardworker_indian wrote: How many integers between 0 and 1570 have a prime tens digit and a prime units digit? (A) 295 (B) 252 (C) 236 (D) 96 (E) 76 There are 4 single digit prime numbers: 2, 3, 5 and 7. Hence, last two digits (tens and units) can take 4*4=16 different values: 22, 23, ..., 77. So, in each hundred there are 16 such numbers. In 16 hundreds there will be 16*16=256 such numbers, but 4 out of them will be more than 1570, namely: 1572, 1573, 1575 and 1577. Which means that there are 2564=252 numbers between 0 and 1570 which have a prime tens digit and a prime units digit. Answer: B. Hope it's clear.
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B) 252  1 min
2357 are primes
4*4=16 possible double digits(units+tens) in first 100 numbers
Since there are 15 hundreds, multiply that by 15 and get 240
For the numbers above 1500, just remove the possibilities of having units+tens prime (72737577) so 164=12
Add up and get 240+12=252
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I certainly misunderstood the question : to me it is to count integers from 0 to 1570 with 1 prime tens digit or 1 prime units digit (nothing to do with this but makes me remember a possible mistake in the 8th challenge...)
from 0 > 999
tens digit prime : 10.4.6 = 240
units digit prime : 10.6.4 = 240
from 1000 > 1570
tens digit prime : 6.4.6 = 144
units digit prime : 6.4.4 = 96
Total : 720 !! Where is the mistake ??



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You have tens and units digits TU
T could be 2357
U could be 2357
Because you need both digits being prime, total possible combinations: 4*4=16
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Yep, I got 252
Prime are 2357
begin with 2 digits number : 16 numbers
then 3 digits number : 16*9 ways numbers
then 4 digits number until 1500 (but we know that the number can not begin by 2, 3, 4, ... because the biggest number is 1570 so we just need to consider 1 possibility for 4 digits) : 16*5 numbers
then 1500 > 1571 : 12 numbers (222325273233353752535557)
total is 16*15 + 12 = 252
what is the OA please



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Re: How many integers between 0 and 1570 have a prime tens digit
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15 Oct 2013, 13:49
Bunuel wrote: hardworker_indian wrote: How many integers between 0 and 1570 have a prime tens digit and a prime units digit? (A) 295 (B) 252 (C) 236 (D) 96 (E) 76 There are 4 single digit prime numbers: 2, 3, 5 and 7. Hence, last two digits (tens and units) can take 4*4=16 different values: 22, 23, ..., 77. So, in each hundred there are 16 such numbers. In 16 hundreds there will be 16*16=256 such numbers, but 4 out of them will be more than 1570, namely: 1572, 1573, 1575 and 1577. Which means that there are 2564=252 numbers between 0 and 1570 which have a prime tens digit and a prime units digit. Answer: A. Hope it's clear. Similar questions to practice: howmanyintegersbetween324700and458600havetens110744.htmlhowmanynumbersbetween0and1670haveaprimetensdigit127319.html
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Re: How many integers between 0 and 1570 have a prime tens digit
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19 Apr 2014, 18:00
Tried to solve it and can't quite understand the following:
you have the 4 digits: _ _ _ _ From right to left: 1*15*4*4 (since units and tens can take either four options: 2,3,5,7) and then you have 15 possibilities (from 1 to 15) to account for hundreds.
16*15 = 240
Yet, I don't get the part where you have to add up those 12 to reach 252, since you already took 15 into account (1500's) and those 16 for the 1500's are also multiplied (1500s + 16 options of tens and hundreds).
What I mean is: It seems to me that those 12 should not be added up since they're implied in the multiplication.
Please help with the flaw in my logic.



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Re: How many integers between 0 and 1570 have a prime tens digit
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20 Apr 2014, 02:50
Enael wrote: Tried to solve it and can't quite understand the following:
you have the 4 digits: _ _ _ _ From right to left: 1*15*4*4 (since units and tens can take either four options: 2,3,5,7) and then you have 15 possibilities (from 1 to 15) to account for hundreds.
16*15 = 240
Yet, I don't get the part where you have to add up those 12 to reach 252, since you already took 15 into account (1500's) and those 16 for the 1500's are also multiplied (1500s + 16 options of tens and hundreds).
What I mean is: It seems to me that those 12 should not be added up since they're implied in the multiplication.
Please help with the flaw in my logic. First of all you might find helpful to go through this: howmanyintegersbetween0and1570haveaprimetensdigit10443.html#p1278818. Also, check similar questions which are given here: howmanyintegersbetween0and1570haveaprimetensdigit10443.html#p1278820As for your solution: 1570 is 15 "complete" hundreds and 70. The same way as 170 is 1 hundred and 70. So, 15*16 is not correct. Next, we are doing 16*16  4: in each hundred there are 16 numbers, which have a prime tens digit and a prime units digit. So, in 16 hundreds there will be 16*16 such numbers. But the last "incomplete" hundred will not have 16 numbers, it will have only 164=12, because 1572, 1573, 1575 and 1577 are greater than 1570. Hope it's clear.
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Re: How many integers between 0 and 1570 have a prime tens digit
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20 Apr 2014, 06:27
I tried solving it but it didn't come out right. Can someone see where the problem is with the logic?
For each of the 1's and 10's digits, there can be 4 options. so we have 16. For the hundreds we have 6 options. For the thousands we have 2 options. So it comes to 12*16 = 192. > wrong!!
Can someone see the problem?



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Re: How many integers between 0 and 1570 have a prime tens digit
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20 Apr 2014, 11:40
ronr34 wrote: I tried solving it but it didn't come out right. Can someone see where the problem is with the logic?
For each of the 1's and 10's digits, there can be 4 options. so we have 16. For the hundreds we have 6 options. For the thousands we have 2 options. So it comes to 12*16 = 192. > wrong!!
Can someone see the problem? If the thousands digit is 0, then for hundreds we have 10 options not 6.
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Re: How many integers between 0 and 1570 have a prime tens digit
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29 Apr 2014, 23:24
Hello there
I tried using the following approach. I did end up taking some time.
The numbers to be considered are 2,3,5 & 7.
For 2digit numbers: (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
4*4 = 16  (1)
For 3digit numbers: (9 numbers in the Hundred's place i.e 1 to 9) (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
9*4*4 = 144  (2)
For 4digit numbers (till 1500): (1 number in the Thousand's place i.e. 1) (5 numbers in the Hundred's place i.e. 0, 1, 2, 3, 4) (4 PRIME numbers in the Ten's place i.e 2, 3, 5 & 7) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*5*4*4 = 80  (3)
For 4digit numbers (1501 to 1570): (1 number in the Thousand's place i.e. 1) (1 number in the Hundred's place i.e. 5) (3 PRIME numbers in the Ten's place i.e 2, 3, 5) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*1*3*4 = 12  (4)
Adding (1), (2), (3) & (4) > 252
Can an Expert evaluate the approach? I want to be sure that I considered the possibilities correctly.
Thanks in advance for the help.



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Re: How many integers between 0 and 1570 have a prime tens digit
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30 Apr 2014, 07:35
shaderon wrote: Hello there
I tried using the following approach. I did end up taking some time.
The numbers to be considered are 2,3,5 & 7.
For 2digit numbers: (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
4*4 = 16  (1)
For 3digit numbers: (9 numbers in the Hundred's place i.e 1 to 9) (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
9*4*4 = 144  (2)
For 4digit numbers (till 1500): (1 number in the Thousand's place i.e. 1) (5 numbers in the Hundred's place i.e. 0, 1, 2, 3, 4) (4 PRIME numbers in the Ten's place i.e 2, 3, 5 & 7) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*5*4*4 = 80  (3)
For 4digit numbers (1501 to 1570): (1 number in the Thousand's place i.e. 1) (1 number in the Hundred's place i.e. 5) (3 PRIME numbers in the Ten's place i.e 2, 3, 5) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*1*3*4 = 12  (4)
Adding (1), (2), (3) & (4) > 252
Can an Expert evaluate the approach? I want to be sure that I considered the possibilities correctly.
Thanks in advance for the help. Yes, your solution is correct.
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Re: How many integers between 0 and 1570 have a prime tens digit
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04 May 2015, 15:03
Bunuel wrote: shaderon wrote: Hello there
I tried using the following approach. I did end up taking some time.
The numbers to be considered are 2,3,5 & 7.
For 2digit numbers: (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
4*4 = 16  (1)
For 3digit numbers: (9 numbers in the Hundred's place i.e 1 to 9) (4 PRIME numbers in the Ten's place) (4 PRIME numbers in the Unit's place)
9*4*4 = 144  (2)
For 4digit numbers (till 1500): (1 number in the Thousand's place i.e. 1) (5 numbers in the Hundred's place i.e. 0, 1, 2, 3, 4) (4 PRIME numbers in the Ten's place i.e 2, 3, 5 & 7) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*5*4*4 = 80  (3)
For 4digit numbers (1501 to 1570): (1 number in the Thousand's place i.e. 1) (1 number in the Hundred's place i.e. 5) (3 PRIME numbers in the Ten's place i.e 2, 3, 5) (4 PRIME numbers in the Unit's place i.e 2, 3, 5 & 7)
1*1*3*4 = 12  (4)
Adding (1), (2), (3) & (4) > 252
Can an Expert evaluate the approach? I want to be sure that I considered the possibilities correctly.
Thanks in advance for the help. Yes, your solution is correct. Hello Bunuel and everyone else. I have been reading the answers and Im on the same page as everyone else but I got answer as 256 because 252 +4 , the 4 comes from single digit number 2,3,5, or 7 only. Hmmmm.... where Im going worng here. I would like kindly to ask for explanaion what others think on this issue



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Re: How many integers between 0 and 1570 have a prime tens digit
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04 May 2015, 19:08
Hi kzivrev, The prompt asks for the numbers that have a prime TENS digit AND a prime UNITS digit. The 1digit numbers 2, 3, 5 and 7 do NOT have a TENS digit, so you're not supposed to include them. GMAT assassins aren't born, they're made, Rich
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Re: How many integers between 0 and 1570 have a prime tens digit
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28 Jul 2016, 04:19
Antmavel wrote: Yep, I got 252
Prime are 2357
begin with 2 digits number : 16 numbers then 3 digits number : 16*9 ways numbers then 4 digits number until 1500 (but we know that the number can not begin by 2, 3, 4, ... because the biggest number is 1570 so we just need to consider 1 possibility for 4 digits) : 16*5 numbers then 1500 > 1571 : 12 numbers (222325273233353752535557)
total is 16*15 + 12 = 252
what is the OA please SIR , If you could please tell me why haven't you taken the 4 prime numbers between 09 i.e. 2,3,5,7 under consideration and started with the tens digit. I have done exactly the same only the difference is that I have added the case of a single digit number formed by 2,3,5,7 and an extra 4 cases are coming up.... please let me know about my mistake



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Re: How many integers between 0 and 1570 have a prime tens digit
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28 Jul 2016, 10:49
Hi suramya26, The prompt asks for the numbers that have a prime TENS digit AND a prime UNITS digit. The 1digit numbers 2, 3, 5 and 7 do NOT have a TENS digit, so you're not supposed to include them. GMAT assassins aren't born, they're made, Rich
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How many integers between 0 and 1570 have a prime tens digit
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19 Aug 2016, 15:03
4/10*4/10*1570=251.2 answer b. 252



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Re: How many integers between 0 and 1570 have a prime tens digit
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19 Aug 2016, 15:20
We have 4 primes thats only one digit: 2,3,5&7. For every 100 integers(1100, 101200 etc) we get 4*4=16 numbers that have a prime for tens and a prime for unit. 16*15(15 sets of 100)= 240. 236 is to small and it can at most be 16 more(256) because we do not have a full set of 100 after 1500. Thus answer choice B is the Only fit. Posted from my mobile device
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Re: How many integers between 0 and 1570 have a prime tens digit
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21 Nov 2017, 07:15
hardworker_indian wrote: How many integers between 0 and 1570 have a prime tens digit and a prime units digit?
(A) 295 (B) 252 (C) 236 (D) 96 (E) 76 Prime numbers between 1 and 10 include 2,3,5 and 7. Total number possibilities for two digit integers (between 1 and 100) = 4*4 = 16 Total number possibilities for three digit integers = 9*4*4 = 144 Possibilities for 4 digit integers (between 10001500) = 1*5*4*4 = 80 and between 15001570 = 1*1*3*4 = 12 Total number of possibilities = 16+144+80+12 = 252 (B)




Re: How many integers between 0 and 1570 have a prime tens digit
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