Finding divisors of 2^n and 30

[cragar]

Homework Statement

How many positive divisors does each of the following have?

$$2^n$$ where n is a positive integer.
and 30

The Attempt at a Solution

for 30 i get 2 , 5 , 3 , 10
but my book says 2 ,3 ,5 I don't understand why 10 isn't a divisor.
and for 2^n I am trying to look for a pattern if n=1 i get no divisors
and n=2 i get 1 divisor and n=3 i get 2 divisors so would it be
2^n has n-1 divisors?

 

[tiny-tim]

hi cragar!

(try using the X² tag just above the Reply box, and write "itex" rather than "tex", and it won't keep starting a new line )

How many positive divisors does each of the following have?
…
for 30 i get 2 , 5 , 3 , 10
but my book says 2 ,3 ,5

i suspect that that's just a hint, and they're telling you those are the prime divisors, and leaving you to carry on from there

(btw, you've missed out two more)

and for 2^n I am trying to look for a pattern if n=1 i get no divisors
and n=2 i get 1 divisor and n=3 i get 2 divisors so would it be
2^n has n-1 divisors?

yes

(though you should be able to prove it more rigorously than that! )

 

[cragar]

ok thanks for your post. so would all the divisors of 30 be 1 , 2 ,5,6,10 ,15 . is one a divisor. for $2^n$ to have divisors it has to be a multiple of 2 so would I divide it by 2 and then i would get $2^{n-1}$
then could i say it has n-1 divisors

 

[tiny-tim]

hi cragar!

would all the divisors of 30 be 1 , 2 ,5,6,10 ,15 .

yes (except i don't know whether 1 counts as a divisor )

for $2^n$ to have divisors it has to be a multiple of 2 so would I divide it by 2 and then i would get $2^{n-1}$
then could i say it has n-1 divisors

better would be …

2ⁿ has only one prime divisor, 2 …

so its only divisors are 2^k for 0 < k < n, of which there are n - 1 ​

(and now try a similar proof for 30 )