Question

If an integer n has three unique prime divisors, then it follows that the largest prime divisor of n is less than or equal to cube root of n. True or False?

Answer 1

Answer:

False

Step-by-step explanation:

Lets call the three prime divisors of n p, q, and r, being r the largest, we know:

$n=q * p * r$

Now, if

$q * p < r$

then

$n < r * r = r^2$

So:

$\sqrt{n} < \sqrt{r^2} = r$

Also, for every natural greater than one, we know:

$\sqrt[3]{n}<\sqrt{n}$

so

$\sqrt[3]{n}<\sqrt{n} < r$

from which:

$\sqrt[3]{n} < r$

So, we see, this means the preposition is false, we can find a particular counterexample:

q=2

p=3

p*q = 6

We need to choose a prime greater than 6

r=7

n= 2 * 3 *7 = 42

$\sqrt[3]{42} = 3.4760 < 7$