+1222 A positive integer is called nice if it is a multiple of $6.$
A certain nice positive integer $n$ has exactly $8$ positive divisors. How many prime numbers are divisors of $n?$
+35 First, If a positive integer is divisible by 6, it must be divisible by three and 2, meaning n must be divisible by 3 and 2.
Second, n has exactly 8 positive integers, giving us hints about the prime factorization and their powers.
Here is a pretty neat trick:
for any integer x, you can tell the number of factors by taking each of the prime's exponent adding one to it, and multiplying the whole thing.
ex: 12: 2^2 * 3^1, factor amount: 3 * 2 = 6
8 can be expressed as 1*8, 2*4, 4 *2, and 1*8.
Remember, n has had 3 and 2, so using the clue above, n could be 3^7 * 2^0 ( not possible), 3^1 * 2^3, 3^3 * 2^1, 2^7 * 3^0 ( not possible)
Some are not possible, leaving us with only 3^3 * 2^1 and 3^3 * 2^1, and both of them have only two prime divisors, so the answer is two.
However, I have not proven if we can add more prime factors.
The answer is no, and let's express that as x. If we do add x (x^1 basically), we would end up with: 3^1 * 2^3 * x^1 and 3^3 * 2^1 * x^1
Both are not possible, as if we add 1 to all the exponents that would simply not multiply up to 8, as we added an x^1 term, and if we add one to that, we would get an EXTRA *2, resulting in 16 factors in both cases.
