Various Types of Factors
Even factors:
Even factors are the factors of number, which are divisible by 2.

Example 1: Find the number of even factors of 58800?
Solution: We first factorize 58800.
58800 = 24 315272
In this case we have to find number of even factors, an even factor is divisible by 2 or we can say smallest power of 2 should be 1 not 0.
Hence a factor must have
2(1 or 2 or 3 or 4)  — 4 factors

3(0 or 1 )       — 1+1=2 factors

5(0 or 1 or 2) — 1+2=3 factors

7(0 or 1 or 2) — 1+2=3 factors

Hence total number of even factors = (4)(2)(3)(3) = 72
Hence, number of even factors of a number
$N={{\mathbf{2}}^{\mathbf{p}}}{{\mathbf{a}}^{\mathbf{q}}}{{\mathbf{b}}^{\mathbf{r}}}{{\mathbf{c}}^{\mathbf{s}}}\mathbf{are}\text{ }\mathbf{p}\left( \mathbf{q}\text{ }+\text{ }\mathbf{1} \right)\left( \mathbf{r}\text{ }+\text{ }\mathbf{1} \right)\left( \mathbf{s}\text{ }+\text{ }\mathbf{1} \right)$

Odd factors:
Odd factors are those factors, which are not divisible by 2.

Example 2: Find the number of odd factors of 58800.
Solution: We first factorize 58800.
58800 = 24 315272
Total number of factors of 58800 is 5x2x3x3=90 and in previous example we calculated total number of even factors is 72.
Number of odd factors = Total number of factors – Number of even factors
=90- 72 = 18

Alternate way: Since odd factors should have power of 2 as 0. Hence odd factor must have
2(0 )  ———- 1 factor

3(0 or 1 )     — 1+1=2 factors

5(0 or 1 or 2) — 1+2=3 factors

7(0 or 1 or 2) — 1+2=3 factors

Total number of odd factors = (1 )(2)(3)(3) = 18
Hence, number of odd factors of a number
$N={{\mathbf{2}}^{\mathbf{p}}}{{\mathbf{a}}^{\mathbf{q}}}{{\mathbf{b}}^{\mathbf{r}}}{{\mathbf{c}}^{\mathbf{s}}}\mathbf{are}\text{ }\mathbf{p}\left( \mathbf{q}\text{ }+\text{ }\mathbf{1} \right)\left( \mathbf{r}\text{ }+\text{ }\mathbf{1} \right)\left( \mathbf{s}\text{ }+\text{ }\mathbf{1} \right)$

Perfect square factors:
If a number is perfect square then its prime factors must have even powers.

Example 3: Find the number of factors of 58800 which  are perfect square?
Solution:
We know that for a number to be a perfect square, its factor must have the even number of powers.
We first factorize 58800.
58800 = 24 315272
Hence perfect square factors must have
2(0 or 2 or 4)—– 3 factors

3( 0 )     —–   1  factor

5(0 or 2) ——- 2 factors

7(0 or 2) — 2 factors

Hence, the number of factors which are perfect square are 3x1x2x2=12

Remember
-If number of factors is odd then the number is a perfect square and vice versa  is also true i.e. if a number is a perfect square then number of factors is  odd.
This is because if number is a perfect square then p, q, and r are even and hence
(p + 1) (q + 1) and (r+ 1) are odd and so product of these numbers is also an odd number.
-If number of factors is even then number is not a perfect square.




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