What is the smallest positive integer that has exactly 18 factors?

Answer 1

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Recall that if n = (p1)^(a1) ... (pr)^(ar) is a prime factorization of n with p1, ..., pr being distinct primes, then the number of factors is (a1 + 1) * ... * (ar + 1).

Since we want 18 factors, this leads us to consider

18 = 17 + 1 <---- Factors from p^17

18 = (1 + 1)(8 + 1) <---- Factors from pq^8

18 = (2 + 1)(5 + 1) <---- Factors from p^2 q^5.

18 = (1 + 1)(2 + 1)(2 + 1) <---Factors from pq^2 r^2 [with p, q, r distinct primes].

The smallest positive integer of each kind listed above are (respectively)

2^17 [big!]

2^8 * 3 = 768

2^5 * 3^2 = 288

2^2 * 3^3 * 5 = 540.

Of these numbers, 288 is the smallest.

I hope this helps!

Answer 2

since 1 is not considered to be a factor then 2^18..has 18 factors of 2