# Thread: 4096 has 13 factors WHY

1. ## 4096 has 13 factors WHY

Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.

2. Originally Posted by reagan3nc
Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.

powers of 2 would yield the smallest number. since powers of 1 would only yield 1 and thus would not reach 4096, and powers of 3 (or more, or a product with numbers greater than 2) would be greater than corresponding powers of 2 and so would not be minimum

3. Hello, reagan3nc!

Why is 4096 the smallest number with 13 factors?
What does 2^n have to do with it?
I understand in order to get an odd number of factors you have to use a perfect square.
There is a theorem that might help explain it.

Given: .$\displaystyle N \:=\:p^a\cdot q^b\cdot r^c\,\cdots$ .
(prime factorization)

. . the number of factors of $\displaystyle N$ is: .$\displaystyle d(N) \:=\:(a+1)(b+1)(c+1)\cdots$

. . . .
Add one to each exponent and multiply.

Example: .$\displaystyle N \:=\:6125 \:=\:5^3\cdot7^2$

. . . $\displaystyle d(6125) \:=\:(3+1)(2+1) \:=\:12$ factors.

We want a number $\displaystyle K$ so that: .$\displaystyle d(K) \:=\: 13 \:=\:12 + 1$

Hence, $\displaystyle K$ has a prime factorization with an exponent of 12: .$\displaystyle K \:=\:p^{12}$

. . And the smallest $\displaystyle K$ occurs when $\displaystyle p = 2.$

Got it?

4. Thanks for all your help

5. Originally Posted by reagan3nc
Why is 4096 the smallest number with 13 factors. What does 2^n have to do with it? I understand in order to get an odd number of factors you have to use a perfect square.

Thanks for anything you can add to the picture.
Since any power of 1 will give only 1 as answer. After this number 2 comes.

$\displaystyle 4096=2^{12}$

$\displaystyle 4096=1.2.2.2.2.2.2.2.2.2.2.2.2$