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Manager  B
Joined: 16 Jan 2011
Posts: 89
How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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Difficulty:   55% (hard)

Question Stats: 63% (02:00) correct 37% (02:43) wrong based on 424 sessions

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How many numbers between 0 and 1670 have a prime tens digit and a prime units digit?

A. 268
B. 272
C. 202
D. 112
E. 262
Math Expert V
Joined: 02 Sep 2009
Posts: 59055

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9
11
Galiya wrote:
Dear all
help me please to solve the problem below:

How many numbers between 0 and 1670 have a prime tens digit and a prime units digit?
A. 268
B. 272
C. 202
D. 112
E. 262

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 17 hundreds there will be 17*16=272 such numbers, but 4 out of them will be more than 1670, namely: 1672, 1673, 1675 and 1677. Which means that there are 272-4=268 numbers between 0 and 1670 which have a prime tens digit and a prime units digit.

Hope it's clear.
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Manager  B
Joined: 16 Jan 2011
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Bunuel
how is it possible to come to number of primes within the hundreds (e.g. 16) quickly? I started to calculate for each ten, then for each hundred and so on... what made me to waste a lot of time on that task
Could you explain ur logic please?
Manager  B
Joined: 31 Jan 2012
Posts: 69
Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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Similar solution as bunuel, but I added. From 1-9 there are 4 prime numbers [2.3.4.7]. The 10s and 1s of the number must be made with these number. Total number of combination is 4*4 = 16. You can arrange the hundreds and thousand numbers in 15 ways [0XX,1XX,2XX..10XX,11XX...15XX]. 16*16= 256. For 1600 it will only go up to 70, so in the 10s column we can only use 3 numbers [2.3.5]. 3*4 =12.

256+12 = 268

Math Expert V
Joined: 02 Sep 2009
Posts: 59055

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Galiya wrote:
Bunuel
how is it possible to come to number of primes within the hundreds (e.g. 16) quickly? I started to calculate for each ten, then for each hundred and so on... what made me to waste a lot of time on that task
Could you explain ur logic please?

Fact that the last two digits (tens and units) can take 4*4=16 different values: 22, 23, 25, ..., 77, basically means that there are 16 such numbers between 0 and 100, the same exact two-digit numbers as listed previously: 22, 23, 25, ..., 77. Thus each hundred will also have 16 such numbers: 122, 123, 125, ..., 177 (for second hundred), 222, 223, 225, ..., 277 (for second hundred), and so on.

Hope it's clear.
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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4
Galiya wrote:
Dear all
help me please to solve the problem below:

How many numbers between 0 and 1670 have a prime tens digit and a prime units digit?
A. 268
B. 272
C. 202
D. 112
E. 262

We have 10 digits (0 - 9) and 4 of them are prime (2, 3, 5, 7). So (4/10)th of consecutive numbers will have prime digits in their unit's place.
Similarly, if we want prime digits in ten's place, again (4/10)th of 100 consecutive numbers will have prime digits in ten's place.

So in 100 consecutive numbers, (4/10)*(4/10) = 16/100 will have prime digits in both unit's and ten's places.

Number of numbers with both units and tens digit prime in the first 1600 = (16/100) * 1600 = 256
Number of numbers with both units and tens digit prime in leftover 70 numbers = (3/4) * 16 = 12

Required number of numbers = 256 + 12 = 268
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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I did kys123 way. only error kys bet 1-9 primes are (2,3,5,7) not 4 (silly mistake). Karishma, did not get that 3/4*16 part. cd you explain me...
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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2
1
sdas wrote:
I did kys123 way. only error kys bet 1-9 primes are (2,3,5,7) not 4 (silly mistake). Karishma, did not get that 3/4*16 part. cd you explain me...

Since we are taking into account only 70 numbers (from 1601 to 1670) and not 100, we will get 3 prime digits in the tens place (2, 3 and 5). We will not get numbers related to 7 in the tens place. So out of four prime digits in the tens place, only three will be used. Therefore, the number of numbers we will get = (3/4)th of 16
(since every prime digit in the tens place will give us the same no. of numbers where the tens and the units place are prime)
22 32 52 72
23 33 53 73
25 35 55 75
27 37 57 77
The fourth column will not be available.
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1
10-99 -> 4*4
100-999 -> 9*4*4
1000-1670 -> 1*7*4*4 - 4

Total = 268
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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GMAT 1: 730 Q50 V39 GRE 1: Q800 V680 Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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1
There are 16 different possible combinations when we consider only the units and tens digits (4 possible prime values for each digit). The digit in the thousands place can be 0 or 1 - when it is 0 there are 10 different possible values for the digit in the hundreds place (0 thru 9) and when the digit in the thousands place is 1 there are 7 different values possible for the hundreds digit (0 thru 6). Thus giving us a total of 16*10+16*7 =272. But of these 4 are >1670, so subtracting, final answer is 268.
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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2
BUNUEL's and KARISHMA's explanations are indeed smart and excellent. With conventional approach,which I adopted, it would take more than 2 minutes to solve.

2,3,5,7 are primes.

From 0 to 1670 there can be 2 digit numbers, 3 digit numbers, and 4 digit numbers (We will not consider 1 digit numbers as we are asked for last two digits of the number)

2 digit numbers = (Any one of 4 primes) * (Any one of 4 primes) = 4 * 4 = 16
3 digit numbers = (Any one from 1 to 9)*(Any one of 4 primes) * (Any one of 4 primes) = 9*4*4 = 144
4 digit numbers(upto 1599) = (Only one digit i.e.1)*(Any one from 0 to 5)*(Any one of 4 primes) * (Any one of 4 primes) = 1*6*4*4 = 96
Numbers between 1600 and 1670 = Here we can see from 2,3,5,and7 we can take any one from 2,3,and5 at tens place and any one of 4 at units place. So Number of numbers = 3*4 = 12

Total Number of Numbers = 16 + 144 + 96 + 12 = 268
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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Why are we not considering single digit prime numbers i.e. 2,3,5,7.

Math Expert V
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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vikrantgulia wrote:
Why are we not considering single digit prime numbers i.e. 2,3,5,7.

Do 2, 3, 5 and 7 have a prime tens digit and a prime units digit? Do a single digit primes even have a tens digit?
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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Way I did it

Prime #'s digits: 2,3,5,7

First , two digit numbers

(4)(4) = 16

Then, 3 digit numbers

(9)(4)(4) = 144

After, 4 digit numbers up to 1600

(1)(6)(4)(496

Finally, from 1600 till 1670

We will need two digit numbers with primes less than 70, meaning

(3)(4) (Excluding 7 from tens digit) = 12

Hope it helps
Cheers!
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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2
Responding to a pm:
Quote:

at first i have arrived at wrong answer . could you please explain what is wrong in my approach.

there are 4 terms so

first term either 0 or 1 - 2
second term from 0 to 6 - 7
third term 2,35,7 ----------4
fourth term same --- 4

finally i got 2*7*4*4 = 224..

are there any way to correct this.. where am i missing the count.

I immediately saw some extra counting in your numbers but I saw that your total is less than the actual. Then I realized that there are a whole bunch of numbers you haven't counted because you are giving values of only 0 to 6 to the hundreds digit.

722, 723, 725, 727, 732... (16 numbers)
822, 823, 825, ... (16 numbers)
922, 923, 925, ... (16 numbers)

So add 16*3 = 48 numbers to your total and subtract 4 (think why - look at the extreme end of your range) to get 224 + 48 - 4 = 268
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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1670*4/10*4/10=267.2
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Re: How many numbers between 0 and 1670 have a prime tens digit  [#permalink]

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XY x and y can take only 4 values so combinations = 4*4=16
For three digits XYZ , Y Z have to be prime so X can take 9 values Y and Z will each be 4 .
So combinations = 9*4*4

For four digits PQRS, P can only be 1 , Q can take only 7 values ( 0-6 ) R and S can each take 4 digits so combinations = 1*7*4*4
In this case we have included 1672 1673 1675 and 1677. Subtract 4.

Final ans = 16+144+112 - 4 = 268.

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