Let-N-be-a-positive-integer-with-prime-factorisation-N-p-1-m-1-p-2-m-2-p-3-m-3-p-n-1-m-n-1-p-n-m-n-where-n-m-i-Z-and-p-r-is-prime-How-many-proper-factors-does-N-have-Inv

Question Number 1942 by Yozzi last updated on 25/Oct/15

L e t N b e a p o s i t i v e i n t e g e r w i t h p r i m e

f a c t o r i s a t i o n

N = p_{1}^{m_{1}} p_{2}^{m_{2}} p_{3}^{m_{3}} \times \dots \times p_{n - 1}^{m_{n - 1}} p_{n}^{m_{n}}

w h e r e n, m_{i} \in Z^{+} a n d p_{r} i s p r i m e .

H o w m a n y p r o p e r f a c t o r s d o e s N h a v e ?

I n v e s t i g a t e c a s e s w h e r e n = 1, n = 2, n = 3

a n d n = 4 . W h a t i s t h e s m a l l e s t p o s i t i v e

i n t e g e r w i t h 12 p r o p e r f a c t o r s ?

W h a t i s t h e s m a l l e s t p o s i t i v e i n t e g e r

w i t h a t l e a s t 12 p r o p e r f a c t o r s ?

(A p r o p e r f a c t o r o f a p o s i t i v e n u m b e r N

i s p o s i t i v e n t e g e r M s u c h t h a t M \neq 1 a n d M \neq N .)

Commented by prakash jain last updated on 25/Oct/15

Total factors, K = \underset{i = 1}{\prod^{n}} (m_{i} + 1)

proper factors = K - 2

Answered by prakash jain last updated on 25/Oct/15

12 proper factor \Rightarrow 14 total factors

5 \times 3 = 15 > 14

m_{1} = 4, m_{2} = 2

Smallest integer = 2^{4} \times 3^{2} = 16 \times 9 = 144