the number of prime numbers less than 1 lakh whose digital sum is 2(digital sum ofanumberisthesumofitsdigits)?

Question

a)5
b)4
c)3
d)none of these

Answer 1

c) 3.

But wait, my reasoning for this choice is different from others. You see, that most people say that the prime nos. less than 1 lakh (or 100 thousand) whose digital sum is 2 are 11,101 and 10001.

But here they are wrong. 10001 is a composite no. and not a prime no.

*************The actual prime nos. less than 100,000 whose digital sum is 2 are:-
2 , 11 , 101.******************

I have tried extensively to find them and check whether 10001 is a prime or not.
The factors of 10001 are 73 and 137, thus 10001 is not prime.

I have also double-checked my answer with a computer. I used Visual Basic code for this purpose. If you are interested in the code, visit:

http://www.geocities.com/anandbrar/code_2006_08.txt

Answer 2

Don't know what you mean by 1 lakh.

As for prime numbers with digital sum 2: we have 2, of course, and any prime number of the form

1 1

(There can be no zeroes behind the last 1, because then you would have a multiple of 10.)

In other words, we look for prime numbers of the form 10^x + 1.

If x is odd, then (10^x + 1) = (-1)^x + 1 = 0 mod 11, so the number would be a multiple of 11. (11 itself is okay, but 1001, 100001, etc. are not.)

If x is a multiple of 3, then
(10^x + 1) = (1000^y + 1) = (-1)^y + 1 = 0 mod 7, so the number would be a multiple of 7. (This rules out 1001, 1000001, etc.)

This reduces the possibilities to 11 and numbers of the form

10^(6x +- 2) + 1

the first of which are 10001, 100000001.

Answer 3

I take it that 1 lakh is equal to100 thousand? In that case, the answer is a i.e., 3.
11, 101, 10001.

Answer 4

Write a program and find out.