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

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  • kb
    Lv 7
    8 years ago
    Favorite Answer

    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!

  • ted s
    Lv 7
    8 years ago

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

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