[xamdam]

How many prime numbers
that are less than a billion
have a digit sum
that is a prime number
that is less than
the sum of the digits
of the number of factors
of a googol?

This post has been edited by xamdam: 07 November 2009 - 07:43 AM

[dummyrummy]

wow.
i am a bit drunk so shall get back to you with the correcto answer shortly

[psychic_mind]

Spoiler for All I can find

3 - (2,3,101)

Number of factors in a googol... A googol is 10¹⁰⁰. 1 is a factor. Every power of 10 up to 100 is also a factor. For each power of 10 there is another power of 2 and 5.
Therefore I count 301 factors. The digit sum of this is 4. So the prime number that is the digit sum of our number must either be 3 or 2. If the digit sum was 3 then it wouldn't be prime with the exception of 3 itself. So we are looking for numbers less than a billion that have a digit sum of 2. It cannot be even , except 2, so it must end and start in a 1. 101 is the only such number less than a billion that is prime.

[txmom2]

Spoiler for it took me a few tries to figure out what you were saying, but...

if 101 is less than a billion and equals two than wouldn't 1001, 10001, 100001....?

psychic_mind, on 07 November 2009 - 06:58 AM, said:

Spoiler for All I can find

3 - (2,3,101)

Number of factors in a googol... A googol is 10¹⁰⁰. 1 is a factor. Every power of 10 up to 100 is also a factor. For each power of 10 there is another power of 2 and 5.
Therefore I count 301 factors. The digit sum of this is 4. So the prime number that is the digit sum of our number must either be 3 or 2. If the digit sum was 3 then it wouldn't be prime with the exception of 3 itself. So we are looking for numbers less than a billion that have a digit sum of 2. It cannot be even , except 2, so it must end and start in a 1. 101 is the only such number less than a billion that is prime.

This post has been edited by txmom2: 07 November 2009 - 01:48 PM

[Tuckleton]

psychic_mind, on 07 November 2009 - 12:58 PM, said:

Spoiler for All I can find

3 - (2,3,101)

Number of factors in a googol... A googol is 10¹⁰⁰. 1 is a factor. Every power of 10 up to 100 is also a factor. For each power of 10 there is another power of 2 and 5.
Therefore I count 301 factors. The digit sum of this is 4. So the prime number that is the digit sum of our number must either be 3 or 2. If the digit sum was 3 then it wouldn't be prime with the exception of 3 itself. So we are looking for numbers less than a billion that have a digit sum of 2. It cannot be even , except 2, so it must end and start in a 1. 101 is the only such number less than a billion that is prime.

Spoiler for I don't disagree with your answer but

I think I disagree with the number of factors you found for a googol. For example, does your analysis include 20 as a factor? I count 10201 factors.

[psychic_mind]

Spoiler for Revised answer

10201 factors. But the rest of my answer should still be correct.

[Zachs]

psychic_mind, on 07 November 2009 - 06:58 AM, said:

Spoiler for All I can find

3 - (2,3,101)

Number of factors in a googol... A googol is 10¹⁰⁰. 1 is a factor. Every power of 10 up to 100 is also a factor. For each power of 10 there is another power of 2 and 5.
Therefore I count 301 factors. The digit sum of this is 4. So the prime number that is the digit sum of our number must either be 3 or 2. If the digit sum was 3 then it wouldn't be prime with the exception of 3 itself. So we are looking for numbers less than a billion that have a digit sum of 2. It cannot be even , except 2, so it must end and start in a 1. 101 is the only such number less than a billion that is prime.

Spoiler for What about...

11? Prime and sum is 2

txmom2, on 07 November 2009 - 07:46 AM, said:

Spoiler for it took me a few tries to figure out what you were saying, but...

if 101 is less than a billion and equals two than wouldn't 1001, 10001, 100001....?

Those numbers aren't prime?

[xamdam]

Nice teamwork, all. Glad you caught that last one, Zachs -- I was afraid it was going to get away. Welcome to the Den!

Spoiler for I counted

four primes: 2, 3, 11 and 101 that had prime digit sums less than the sum of the digits of 10201

This post has been edited by xamdam: 10 November 2009 - 02:26 AM

[Zachs]

xamdam, on 09 November 2009 - 08:20 PM, said:

Nice teamwork, all. Glad you caught that last one, Zachs -- I was afraid it was going to get away. Welcome to the Den!

Spoiler for I counted

four primes: 2, 3, 11 and 101 that had prime digit sums less than the sum of the digits of 10201

Thanks. My friend and I have been doing these for a while now. I try to figure out the 'trick' to them. He wrote a custom data type and brute forces everything on his computer. We couldn't let that one slip through though. :-)

[xamdam]

Zachs, on 12 November 2009 - 03:33 AM, said:

Thanks. My friend and I have been doing these for a while now. I try to figure out the 'trick' to them. He wrote a custom data type and brute forces everything on his computer. We couldn't let that one slip through though. :-)

Sounds like a good combination.

Just curious -- there was a "Factors and Factorials" puzzle a few weeks ago that contained numbers like 9001 factorial. Can your friend's custom data type handle numbers that big?