Numbers Less Than 100 With Eight Divisors
\[n=p_1^{k_1}p_2^{k_2}...p_r^{k_r}\]
is written \[d(n)\]
and is equal to \[d(n)=d(p_1^{k_1}p_2^{k_2}...p_r^{k_r})=(k_1+1)(k_2+1)...(k_r+1)\]
.If
\[n\]
has eight divisors then the prime powers must be 1-1=0, 2-1=1, 4-1=3, or 8-1=7.Since
\[2^7=128 \gt 100\]
there is no number of the form \[p^7 \lt 100\]
. The only forms of a number less than 100 with eight divisors are \[p^3r\]
and \[pqr\]
. They are\[2^3 \times 3=24\]
\[2^3 \times 5=40\]
\[2^3 \times 7=56\]
\[2^3 \times 11=88\]
\[3^3 \times 2=54\]
\[2 \times 3 \times 5=30\]
\[2 \times 3 \times 7=42\]
\[2 \times 3 \times 11=66\]
\[2 \times 3 \times 13=78\]
\[2 \times 5 \times 7=70\]
