selango wrote:1.If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

2.If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

Are the above 2 properties correct?

1.If N is a positive integer which is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

This is always true.

2.If N is a positive integer which is NOT a perfect square, then the number of factors of N will ALWAYS be an EVEN number.

This is again always true.

what does perfect square and non perfect square means


25,36,49 theses are perfect?

pradeepkaushal9518 wrote:what does perfect square and non perfect square means


25,36,49 theses are perfect?

Yes.. 25,36,49.. are all perfect squares, because they are squares of a positive integer.

2 is a non perfect square. because its root is irrational.
root(2) = 1.4142...

Hope this helps!!

Thanks..I got confused that number of distinct factors is Odd.

kvcpk wrote:

pradeepkaushal9518 wrote:what does perfect square and non perfect square means


25,36,49 theses are perfect?

Yes.. 25,36,49.. are all perfect squares, because they are squares of a positive integer.

2 is a non perfect square. because its root is irrational.
root(2) = 1.4142...

Hope this helps!!

25,36,49.. are all perfect squares, because they are squares of an integer, which could be negative too. In fact, a perfect square number is one whose square root is a rational number, positive or negative. But when we talk about factors of N, N is got to be a positive integer only.

selango wrote:1.If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

2.If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

Are the above 2 properties correct?

Yes, as a matter of fact, we can remember the following rule:
If a number N can be expressed as a product of powers of primes, i.e. N = (a^x) * (b^y)*(c^z)...., then the number of factors of N is given by (x+1) * (y+1) * (z+1)....
e.g since 210 = 2*5*3*7, the number of factors is (1+1)*(1+1)*(1+1)*(1+1) = 16. This formula is very easy to remember and derive.

Now since perfect squares will always have even numbers as powers of primes, the number of factors, as given by the formula, will always be odd because we are adding 1 to each even number. Conversely for numbers that are not perfect squares, the number of factors is always even.

Vipulvp wrote:

selango wrote:1.If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

2.If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

Are the above 2 properties correct?

Yes, as a matter of fact, we can remember the following rule:
If a number N can be expressed as a product of powers of primes, i.e. N = (a^x) * (b^y)*(c^z)...., then the number of factors of N is given by (x+1) * (y+1) * (z+1)....
e.g since 210 = 2*5*3*7, the number of factors is (1+1)*(1+1)*(1+1)*(1+1) = 16. This formula is very easy to remember and derive.

Now since perfect squares will always have even numbers as powers of primes, the number of factors, as given by the formula, will always be odd because we are adding 1 to each even number. Conversely for numbers that are not perfect squares, the number of factors is always even.

That's excellent

selango wrote:1.If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

2.If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

Are the above 2 properties correct?

Let's take it this way:

1. Since N is a perfect square take any perfect square number (say 4)

If we find the factors of 4 we get 1,2 and 4

Thus from the above observation we get that there are 3 factors.

Thus statement 1 is correct. A perfect square has odd number of factors.


2. This one you can take as root 2 and proceed as shown.

sanju09 wrote:

selango wrote:1.If N is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

2.If N is a NON-perfect square, then the number of factors of N will ALWAYS be an EVEN number.

Are the above 2 properties correct?

1.If N is a positive integer which is a perfect square, then the number of factors of N will ALWAYS be an ODD number.

This is always true.

2.If N is a positive integer which is NOT a perfect square, then the number of factors of N will ALWAYS be an EVEN number.

This is again always true.

Thank you sanju09.The question is bit of confused. Now I got clear idea.