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

Question

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

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 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. Answer: B. Similar questions to practice: https://gmatclub.com/forum/how-many-inte ... 10744.html https://gmatclub.com/forum/how-many-numb ... 27319.html Hope it's clear.

Paul · Answer

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
Add up and get 240+12=252

Antmavel · Answer

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

what is the OA please

Enael · Answer

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.

Bunuel · Answer

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.

shaderon · Answer

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.

Bunuel · Answer

Yes, your solution is correct.

EMPOWERgmatRichC · Answer

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.

GMAT assassins aren't born, they're made,
Rich

suramya26 · Answer

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....
please let me know about my mistake

EMPOWERgmatRichC · Answer

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,
Rich

JS1290 · Answer

Hi  Bunuel ,

I have followed a similar approach; however, I got a different answer. For 2-digit numbers, I got 16. For 3-digit numbers, I got 144. For 4-digit numbers, I had a different approach. My approach was to find integers from 1000 to 1569 for the 4 digit numbers. 1*6*4*4. First field: We only have one way to pick digit 1. Second field, we have 6 different ways to pick a digit between 0 and 5. Third & fourth field: we have 4 different ways of picking a prime number. Could you please explain where I went wrong in my approach? Would greatly appreciate it!

EMPOWERgmatRichC · Answer

Hi csaluja,

Your calculation includes 4 numbers that are GREATER than 1570 (re: 1572, 1573, 1575 and 1577). Those numbers are outside of the range defined by the question, so they should not be included.

GMAT assassins aren't born, they're made,
Rich

ScottTargetTestPrep · Answer

Since 4 digits are prime digits (2, 3, 5, 7), if the number has 2 digits, then we have 4 x 4 = 16 such numbers. If the number has 3 digits, then we have 9 x 4 x 4 = 144 such numbers. If the number has 4 digits and it’s less than 1500, then we have 1 x 5 x 4 x 4 = 80 such numbers. Finally, if the number has 4 digits and it’s between 1500 and 1570 (inclusive), then we have 1 x 1 x 3 x 4 = 12 such numbers. Therefore, we have a total of 16 + 144 + 80 + 12 = 252 numbers between 0 and 1570 that have a prime tens digit and a prime units digit.

Answer: B

IanStewart · Answer

From 1 through 1500, exactly 2/5 of all whole numbers have a prime units digit, and 2/5 have a prime tens digit. Digits are independent, so (2/5)(2/5) = 16% of the whole numbers up to 1500 have prime tens and units digits, or 240 of them. The answer is slightly larger than that, because some numbers between 1501 and 1570 should also be counted, but glancing at the answer choices, only 252 makes any sense.

kajaldaryani46 · Answer

—-
I used the same approach to solve but I think what I missed doing was breaking up the 4 digit nos into 2 parts - one less than 1500 and the other between 1500-1570.

So, I calculated the 4 digits nos that satisfy the criteria using the logic below:
1(1st digit which will be 1)* 6(ways to select from 0-5)* 3(7 cant be selected here from the 4 prime nos as the no will go beyond 1570)*4(any of the 4 prime nos can be selected).

I’m not sure why this approach gives me a different ans than the one that you explained above. I’m missing some cases clearly but why is that happening because I think I have accounted for the criteria appropriately?

It’d be great if somebody can help me understand.  Bunuel  would appreciate your inputs!

Thanks

Posted from my mobile device

IanStewart · Answer

Yes, you're right about what you need to do to correct this solution. When you're counting your choices for each digit, if you say "there are 6 choices for the hundreds digit, 0, 1, 2, 3, 4 or 5", then when you come to the tens digit, you're stuck. *If* we chose "5" for the hundreds digit, we have three choices for the tens digit (2, 3 or 5), because the number overall is less than 1570. But *if* we chose 0, 1, 2, 3 or 4 for the hundreds digit, we have four choices for the tens digit (we can use any prime, 2, 3, 5 or 7). Any time, when counting, you discover your later choices depend on which choices you made earlier, you need to either solve differently or divide the problem into cases. That's what's true here -- we don't know how many choices we have for the tens digit because it depends on what we chose for the hundreds digit.

Because you disallowed the selection of "7" in the tens place altogether, you haven't counted a lot of numbers that should be counted, like 1372 or 1277. If you divide the problem into cases (either by considering numbers from 1500-1570 separately, or by considering numbers with a tens digit of "7" separately) your approach should lead to a correct answer.

Abhishekde · Answer

I get the 252. But we need to subtract the numbers 2,3,5,&7 from here. because the tens digit of these number (between0-10) is 0 which is not a prime. S0 252-5=248

Bunuel · Answer

Nope. The solution you quote does not mention that 2, 3, 5, and 7 are included in 252. We are including only the numbers ending in two primes: 22, 23, ..., 77. So, the answer is 252.

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

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

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