daretodream wrote:Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even

A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Using the above facts, you can conclude that both statements are sufficient to answer the question.

thephoenix wrote:
A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Are these statements always true...They are very helpful .I am going to add these on my flashcards..
can you give some link where I can find these and similar ones for odd and even number properties...

shashank.ism wrote:

thephoenix wrote:
A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Are these statements always true...They are very helpful .I am going to add these on my flashcards..
can you give some link where I can find these and similar ones for odd and even number properties...

hey shashank even this one was from my flash cards

how ever i have tried with few examples and tested bth the theorem

A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD

N=4 #of distinct factors are 1,2,4 i.e 3 (odd)
N=9 # of distinct factors are 1,3,9 i.e 3(odd)
N=16 #of distinct factors are 1,2,4,8,16 i.e 5(odd)
N=25 #of distinct factors are 1,5,25 i.e 3(odd)
N=64 #of distinct factors are 1,2,4,8,16,32,64 i.e 7 (odd)
N=81 #of distinct factors are 1,3,9,27,81 i.e 5 (odd)

hence engh to conclude that A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD


II)
if u will luk at abve example u will find that
sum of count of odd factors is odd
and sum of count of evn factors is evn
so we can conclude that
A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors

hth

daretodream wrote:Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even

say a perfect square

P is represented as x^a*y^b*z^c....

Where x,y,z... are prime numbers
then a,b,c should be Positive even integers

the no of factors = (a+1)(b+1)(c+1).... is always odd since (a+1),(b+1),(c+1).... etc all are odd

Sum of factors = (x^(a+1)-1)/(a-1) *(y^b+1)-1/(b-1)*.....

=(1+x+x^2...+x^a)(1+y+y^2+.....+y^a)*....

when a is even all of these factors will be odd and the sum will be odd.

thephoenix wrote:

shashank.ism wrote:

thephoenix wrote:
A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Are these statements always true...They are very helpful .I am going to add these on my flashcards..
can you give some link where I can find these and similar ones for odd and even number properties...

hey shashank even this one was from my flash cards

how ever i have tried with few examples and tested bth the theorem

A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD

N=4 #of distinct factors are 1,2,4 i.e 3 (odd)
N=9 # of distinct factors are 1,3,9 i.e 3(odd)
N=16 #of distinct factors are 1,2,4,8,16 i.e 5(odd)
N=25 #of distinct factors are 1,5,25 i.e 3(odd)
N=64 #of distinct factors are 1,2,4,8,16,32,64 i.e 7 (odd)
N=81 #of distinct factors are 1,3,9,27,81 i.e 5 (odd)

hence engh to conclude that A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD


II)
if u will luk at abve example u will find that
sum of count of odd factors is odd
and sum of count of evn factors is evn
so we can conclude that
A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors

hth

Phoenix i would certainly like to have a look on ur flash card....I think its a source of information now..
will u send me someday..

Well u have done a good experiment for proving both the statement ..