How many factors does thirty billion have?

Answer 1

[I'll assume 30 billion to mean 30 000 000 000, although some would consider it to denote 30 000 000 000 000.] Start with the prime factorisation of 30 000 000 000. 30 000 000 000 = 3 x 1010 = 3 x (2 x 5)10 = 210 x 3 x 510 The factors of 30 billion can be counted using this prime factorisation. First there are the factors of the form 2n, 3n or 5n. For 2 and 5, n varies between 1 and 10. 10 + 10 + 1 = 21 Then there are ten ways of multiplying 3 with 2n, ten more of multiplying 3 with 5n, and 100 ways of multiplying 2n with 3n. 10 + 10 + 100 = 120 Finally, there are 100 ways of multiplying 2n x 3 x 5n, one of which is 30 billion itself. We might also include, trivially, 1. 100 + 1 = 101. That leaves us with 101 + 120 + 21 = 242 factors, including one and 30 billion itself.