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djyliett [7]
3 years ago
6

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 cu

be root of n. True or False?
Mathematics
1 answer:
TiliK225 [7]3 years ago
4 0

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}

so

\sqrt[3]{n}

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

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