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If \(p\) and \(q\) are prime numbers, then the number \(p^a\times q^b\) will have \((a+1)(b+1)\) factors. This is because all its factors are
of the form \(p^\alpha\times q^\beta\), with \(\alpha=0,1,...,a\) and \(\beta=0,1,...,b\). The same idea can be used on numbers with three or more prime factors; in general the number \(p_1^{a_1}\times...\times p_n^{a_n}\) has \((a_1+1)\times...\times(a_n+1)\) factors.
28 can be written as: 28, 14×2, 7×4, or 7×2×2. Therefore the following numbers have 28 factors:
$$2^{27},\quad
2^{13}\times3^1,\quad
2^{6}\times3^3,\quad
2^{6}\times3^1\times5^1
$$
and any other number with 28 factors will have larger prime factors, so will be larger.
These numbers are 134217728, 24576, 1728 and 960. Therefore the smallest number with 28 factors is 960.
