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

Question

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

Bunuel · Accepted Answer

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.

Answer: A.

Hope it's clear.

KarishmaB · Answer

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

galiya · Answer

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?

kys123 · Answer

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

Answer = A

256+12 = 268

Answer = A

Bunuel · Answer

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: 1 22 , 1 23 , 1 25 , ..., 1 77  (for second hundred), 2 22 , 2 23 , 2 25 , ..., 2 77  (for second hundred), and so on.

Hope it's clear.

sdas · Answer

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...

KarishmaB · Answer

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.

iwillcrackgmat · Answer

10-99 -> 4*4
100-999 -> 9*4*4
1000-1670 -> 1*7*4*4 - 4

Total = 268

Bunuel · Answer

Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

Theory on Combinations: math-combinatorics-87345.html

DS questions on Combinations: search.php?search_id=tag&tag_id=31
PS questions on Combinations: search.php?search_id=tag&tag_id=52

Tough and tricky questions on Combinations: hardest-area-questions-probability-and-combinations-101361.html

vs129 · Answer

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.

Narenn · Answer

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

vikrantgulia · Answer

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

Please elaborate

Bunuel · Answer

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?

jlgdr · Answer

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

Now adding up= 144+16+96+12=268

Answer: A

Hope it helps
Cheers!
J

KarishmaB · Answer

Responding to a pm:

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

gracie · Answer

1670*4/10*4/10=267.2
answer a. 268

ShankSouljaBoi · Answer

Start with two digits.
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.

Posted from my mobile device

D4kshGargas · Answer

My math just doesn't add up

0-100 16 numbers

100 - 1000 (16 x 9) numbers

1000 - 1600 (16 x 7) numbers

1600 - 1670 (12) numbers

Therefore total
16(1 + 9 + 7) + 12
16x18 + 12
288 + 12 = 300 such numbers.

Where am I going wrong?

KarishmaB · Answer

The highlighted is incorrect. 
From 1000 to 1599, hundreds digit can take 6 values not 7 (0,1,2,3,4,5).
You are counting the 12 1600 values separately.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards