Advent calendar 2025
10 December
2025 is the smallest number with exactly 15 odd factors.
What is the smallest number with exactly 16 odd factors?
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If the numbers \(p_1\), \(p_2\), ..., \(p_n\) are odd prime numbers and \(i_1\), \(i_2\), ..., \(i_n\) are positive integers, then the number \(p_1^{i_1}p_2^{i_2}...p_n^{i_n}\)
has \((i_1+1)(i_2+1)...(i_n+1)\) odd factors.
If \((i_1+1)(i_2+1)...(i_n+1)=16\) the the possible values for the \(i\)s are:
- 15
- 7 and 1
- 3 and 3
- 3, 1 and 1
- 1, 1, 1 and 1
These options lead to:
- 315 = 14348907
- 37×51 = 10925
- 33×53 = 3375
- 33×51×71 = 945
- 31×51×71×111 = 1155
