How many factors does the number 1 2 3 4 5 6 7 8 9 1 0 have, given that the number has only 3 prime factors?
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It is not immediately obvious to see that 1234567891 is prime.
How does the fact there are 3 prime factors help? We don't know the powers of these.
If 1234567891 is not prime, the the number would have 2,4,5, ... prime factors. So 1234567891 has to be prime.
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Well, if 1 2 3 4 5 6 7 8 9 1 was not prime, but the n th power p n of a prime p = 2 , 5 , then the original number would only have three prime factors ( 2 , 5 , p ), but would then have 4 ( n + 1 ) factors overall ( p m , 2 p m , 5 p m , 1 0 p m for 0 ≤ m ≤ n ). You do need to show that 1 2 3 4 5 6 7 8 9 1 is prime (it is).
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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