Factors of a very big number

How many factors does the number 12345678910 12345678910 have, given that the number has only 3 prime factors?

40 12 32 8 20

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1 solution

The prime factorization of 12345678910 is very easy to find.

12345678910=2x5x1234567891

Then, we use the prime factorization the solve the problem.

2x2x2=8.

So the answer is 8 \boxed 8

It is not immediately obvious to see that 1234567891 is prime.

Pi Han Goh - 10 months, 3 weeks ago

How does the fact there are 3 prime factors help? We don't know the powers of these.

Vinayak Srivastava - 10 months, 3 weeks ago

If 1234567891 is not prime, the the number would have 2,4,5, ... prime factors. So 1234567891 has to be prime.

A Former Brilliant Member - 10 months, 3 weeks ago

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Well, if 1234567891 1234567891 was not prime, but the n n th power p n p^n of a prime p 2 , 5 p \neq 2,5 , then the original number would only have three prime factors ( 2 , 5 , p 2,5,p ), but would then have 4 ( n + 1 ) 4(n+1) factors overall ( p m , 2 p m , 5 p m , 10 p m p^m,\,2p^m,\,5p^m,\,10p^m for 0 m n 0 \le m \le n ). You do need to show that 1234567891 1234567891 is prime (it is).

Mark Hennings - 8 months, 2 weeks ago

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