If-the-number-of-divisors-of-a-number-is-odd-prove-that-the-number-is-perfect-square-and-vice-versa-

Question Number 27767 by Rasheed.Sindhi last updated on 14/Jan/18

If the number of divisors of a

number is odd, prove that the

number is perfect square and

vice versa .

Answered by mrW2 last updated on 14/Jan/18

E v e r y n u m b e r n c a n b e e x p r e s s e d a s

n = \underset{i = 1}{\prod^{k}} p_{i}^{e_{i}}, w h e r e p_{i} i s p r i m e,

a n d t h e n u m b e r o f i t s d i v i s o r s i s

d (n) = \underset{i = 1}{\prod^{k}} (e_{i} + 1)

I f a n u m b e r N i s a p e r f e c t s q u a r e,

i . e . N = n^{2}, t h e n

N = {(\underset{i = 1}{\prod^{k}} p_{i}^{e_{i}})}^{2} = \underset{i = 1}{\prod^{k}} p_{i}^{2 e_{i}}

d (N) = \underset{i = 1}{\prod^{k}} (2 e_{i} + 1)

s i n c e t h e p r o d u c t o f o d d n u m b e r s i s o d d,

t h e r e f o r e d (N) i s o d d .

I f d (n) = \underset{i = 1}{\prod^{k}} (e_{i} + 1) i s o d d, i t m e a n s e_{i}

m u s t b e e v e n, {l e t}^{'} s s a y e_{i} = 2 f_{i}, t h e n

n = \underset{i = 1}{\prod^{k}} p_{i}^{e_{i}} = \underset{i = 1}{\prod^{k}} p_{i}^{2 f_{i}} = {(\underset{i = 1}{\prod^{k}} p_{i}^{f_{i}})}^{2}

i t m e a n s n i s a p e r f e c t s q u a r e .

Commented by mrW2 last updated on 15/Jan/18

A n o t h e r w a y t o p r o v e :

G e n e r a l l y, i f m i s a d i v i s o r o f t h e

n u m b e r N, t h e n \frac{N}{m} i s a l s o a d i v i s o r

o f i t . T h a t m e a n s t h e d i v i s o r s a r e

a l w a y s p a i r w i s e . E v e r y d i v i s o r h a s

a c o r r e s p o n d i n g p a r t n e r .

I f t h e n u m b e r N i s p e r f e c t s q u a r e, i . e .

N = n^{2}, i t m e a n s n i s a d i v i s o r o f i t,

b u t \frac{N}{n} = n, t h e p a r t n e r o f n i s i n f a c t

i t s e l f, i t m e a n s

i f t h e n u m b e r i s p e r f e c t s q u a r e, i t s

n u m b e r o f d i v i s o r s i s o d d .

O n t h e o t h e r s i d e, i f t h e n u m b e r o f

d i v i s o r s o f N i s o d d, o n e d i v i s o r, s a y n,

a n d i t s p a r t n e r h a s t h e s a m e v a l u e, i . e .

n = \frac{N}{n}

\Rightarrow N = n^{2}

o r N i s p e r f e c t s q u a r e .

Commented by Rasheed.Sindhi last updated on 15/Jan/18

T h a n k s V e r y M u c h S i r!

K n o w l e d g e i n c r e a s i n g a n s w e r f o r m e!