Problem Solving

2008 wrote:

sudhir3127 wrote:30= 2 *3*5

hence its (1+1)*(1+1)*(1+1) number of factors..

thus eight...

maybe i m just burn out this afternoon, but i dont get it... can you please explain this formula?

If you need to count the number of divisors of x:

1) prime factorize x, and write the prime factorization in the conventional notation (using exponents if primes are repeated);
2) now look only at the exponents: add one to each exponent, and multiply the resulting numbers.

To take an example, how many positive divisors does 180 have?

1) Prime factorize: 180 = (2^2)(3^2)(5^1)
2) Add one to each power and multiply: 3*3*2 = 18.

So 180 has 18 different positive divisors. You can see why this works: any number that can be written as (2^a)(3^b)(5^c) will be a divisor of (2^2)(3^2)(5^1) just as long as:

a = 0, 1 or 2 (three choices)
b = 0, 1 or 2 (three choices)
c = 0 or 1 (two choices)

Now it's a counting problem- we multiply the number of choices we have for each of a, b and c: 3*3*2 = 18.