Digit sum of a prime

Number Theory Level 2

How many prime numbers are there such that the sum of their digits is 999?

Also try this harder problem.

Infinitely Many 0 1 999 2

1 solution

Dhruv Bhasin
May 30, 2015

We will use the divisibilty test of 3 which states that if the sum of a number is divisible by 3 then the number is also divible by 3.

Note that $999$ is divisible, therefore, so is every number whose sum of digits is $999$ . This means that no number whose sum of digits is $999$ is a prime, hence the answer is 0.

Can u prove the divisibility test of 3?

Maninder Dhanauta - 6 years ago

This wil be helpful to you.

Dhruv Bhasin - 6 years ago

Yes, the proof is quite easy using Congruences. Just note that $10 \equiv 1 \pmod{3}$ , and then the proof is as simple as anything

Raushan Sharma - 5 years, 8 months ago

very good explanation....

Debmalya Mitra - 6 years ago

Thanks a lot. :)

Dhruv Bhasin - 6 years ago

997+2=999 Man you made my points go down

Amitya Sharma - 6 years ago

Umm...Sorry, but I dont get you?

Dhruv Bhasin - 6 years ago

I got what you meant.

Maybe my reply to Mayur Ade would be helpful to you.

Dhruv Bhasin - 6 years ago

Oh I read question in sort of a hurry now I get It Nice question man and sorry for accusing you

Amitya Sharma - 6 years ago

@Amitya Sharma – No worries friend. Glad you like it. :)

Dhruv Bhasin - 6 years ago

Is 997 a prime number ?..

Mayur Ade - 6 years ago

It indeed is and i believe you are thinking that $997+2=999$ . (And now I also get what Amritya wanted to say.)

But the problem does not demand the sum of different prime numbers but the digut sum of a single prime number. For eg- digit sum of $997$ is $9+9+7=25$ .

Hope that helps.:)

Dhruv Bhasin - 6 years ago

Oh my god, it was so simple and I was creating a program that will count the numbers of prime numbers... I am fool.

Uttam Manher - 2 years, 11 months ago

Digit sum of a prime

Also try this harder problem.

1 solution

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