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

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Senior Manager
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How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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Updated on: 15 Oct 2013, 13:48
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Question Stats:

53% (02:39) correct 47% (02:54) wrong based on 753 sessions

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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

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.
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Posts: 60727
Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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15 Oct 2013, 13:47
15
14
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 256-4=252 numbers between 0 and 1570 which have a prime tens digit and a prime units digit.

Hope it's clear.
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04 Oct 2004, 18:17
11
10
B) 252 - 1 min
2-3-5-7 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 (72-73-75-77) so 16-4=12
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10 Oct 2004, 18:49
3
Yep, I got 252

Prime are 2-3-5-7

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 (22-23-25-27-32-33-35-37-52-53-55-57)

total is 16*15 + 12 = 252

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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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29 Apr 2014, 23:24
2
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 2-digit numbers:
(4 PRIME numbers in the Ten's place)
(4 PRIME numbers in the Unit's place)

4*4 = 16 --- (1)

For 3-digit 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 4-digit 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 4-digit 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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Posts: 3273

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05 Oct 2004, 08:24
1
You have tens and units digits T-U
T could be 2-3-5-7
U could be 2-3-5-7
Because you need both digits being prime, total possible combinations: 4*4=16
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Posts: 60727
Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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15 Oct 2013, 13:49
1
1
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 256-4=252 numbers between 0 and 1570 which have a prime tens digit and a prime units digit.

Hope it's clear.

Similar questions to practice:
how-many-integers-between-324-700-and-458-600-have-tens-110744.html
how-many-numbers-between-0-and-1670-have-a-prime-tens-digit-127319.html
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Posts: 60727
Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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30 Apr 2014, 07:35
1
1
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 2-digit numbers:
(4 PRIME numbers in the Ten's place)
(4 PRIME numbers in the Unit's place)

4*4 = 16 --- (1)

For 3-digit 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 4-digit 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 4-digit 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  [#permalink]

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04 May 2015, 19:08
1
Hi kzivrev,

The prompt asks for the numbers that have a prime TENS digit AND a prime UNITS digit.

The 1-digit numbers 2, 3, 5 and 7 do NOT have a TENS digit, so you're not supposed to include them.

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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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28 Jul 2016, 10:49
1
Hi suramya26,

The prompt asks for the numbers that have a prime TENS digit AND a prime UNITS digit.

The 1-digit 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,
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05 Oct 2004, 02:59
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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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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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.

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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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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.

First of all you might find helpful to go through this: how-many-integers-between-0-and-1570-have-a-prime-tens-digit-10443.html#p1278818. Also, check similar questions which are given here: how-many-integers-between-0-and-1570-have-a-prime-tens-digit-10443.html#p1278820

As 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 16-4=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  [#permalink]

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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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Posts: 60727
Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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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  [#permalink]

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04 May 2015, 15:03
Bunuel 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 2-digit numbers:
(4 PRIME numbers in the Ten's place)
(4 PRIME numbers in the Unit's place)

4*4 = 16 --- (1)

For 3-digit 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 4-digit 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 4-digit 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.

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  [#permalink]

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28 Jul 2016, 04:19
Antmavel wrote:
Yep, I got 252

Prime are 2-3-5-7

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 (22-23-25-27-32-33-35-37-52-53-55-57)

total is 16*15 + 12 = 252

SIR ,
If you could please tell me why haven't you taken the 4 prime numbers between 0-9 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....
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How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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19 Aug 2016, 15:03
4/10*4/10*1570=251.2
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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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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(1-100, 101-200 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.

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Re: How many integers between 0 and 1570 have a prime tens digit  [#permalink]

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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 1000-1500) = 1*5*4*4 = 80
and between 1500-1570 = 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   [#permalink] 21 Nov 2017, 07:15

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