[Guest]

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?

Edited November 7, 2009 by xamdam

[Guest]

wow.

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

[Guest]

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.

[Guest]

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

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.

Edited November 7, 2009 by txmom2

[Guest]

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.

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.

[Guest]

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

[Guest]

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.

11? Prime and sum is 2

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

Those numbers aren't prime?

[Guest]

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

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

Edited November 10, 2009 by xamdam

[Guest]

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

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. :-)

[Guest]

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?