Factors
Number of factors
To find the number of factors of a number, convert the number into the product of its prime factors. Prime factors of a number are the divisors that are prime numbers. Hence 720 can be represented as
720 = 16 * 9 * 5 = $$2^4$$ * $$3^2$$ * 5
The individual factors of 720 can be constructed by taking 2 0 times, 1 time, 2 times . . 4 times = 5 ways
Similarly 3 can be taken 0 times, 1 time or 2 times = 3 ways
Similarly 5 can be taken 0 times or 1 time= 2 ways
Hence the number of factors of 720 are = number of combinations of factors = (4+1) * (2+1) * (1+1) = 5*3*2=30.
In general if the number can be represented as N = $$a^{p}$$ ∗ $$b^{q}$$ ∗ $$c^{r}$$ . . . then number of factors is (p+1) * (q+1) * (r+1) . . .
If the number of factors are odd then N is a perfect square.
If there are n factors, then the number of pairs of factors would be n/2. If N is a perfect square then number of pairs (including the square root) is (n+1)/2.
Number of even and odd factors
To get even factors of 720, we must take 2 as a factor at least once.
Hence 2 can be taken 1 time, 2 times, 3 times, 4 times = 4 ways
If the number can be expressed as N = $$2^{p}$$ ∗ $$a^{q}$$ ∗ $$b^{r}$$ . . . where the power of 2 is p,
Then the number of even factors of N = p (1+q) (1+r) . . .
The number of odd factors of N = (1+q) (1+r) . . .
Hence the number of even factors of 720 (as shown above) = 4 * (2+1) * (1+1) =4 * 3 * 2 = 24
Hence the number of odd factors of 720 (as shown above) = (2+1) * (1+1) = 3 * 2 = 6
Sum of factors
If the number can be represented as N = $$a^{p}$$ \∗ $$b^{r}$$ \∗ $$c^{q}$$ . . .
Then sum of factors is
$$\frac{a^{p+1} -1}{a-1}$$ * $$\frac{b^{q+1} -1}{b-1}$$ * $$\frac{c^{r+1} -1}{c-1}$$ . . .
Hence sum of factors of 720 = $$\frac{2^{5} -1}{2-1}$$ * $$\frac{3^{2+1} -1}{3-1}$$ * $$\frac{5^{1+1} -1}{5-1}$$ = 31 * 13 * 6 = 2418
