YLL討論網

n是一個自然數,2n有28個因數,3n有30個因數,那麼6n有多少個因數?

Let n = 2^a1 * 3^a2 *.... pn^an
2n = 2^(a1+1)...
No. of factors = (a1+2)(a2+1)...(an+1)=28
3n = 2^a1 * 3^(a2+1)...
No. of factors = (a1+1)(a2+2)...(an+1)=30

28 = 2*2*7
a1+2 = 4 or 7 or 14
a1= 2 or 5 or 12
Since a1+1|30
a1+1=3 or 6 or 13
a1 = 2 or 5
If a1=2
a2+1=7
a2=6
a2+2 is not divisible by 30-->rejected
If a1=5
a2+1=2 or 4
a2 = 1 or 3
a2+2 = 3 or 5
Since a1=5
30/(5+1)=5
Therefore a2=3
So n = 2^ 5 * 3^3
6n = 2^6 * 3^4
No. of factors: (6+1)(4+1) = 35

1000