Question

Show that if the smallest prime factor pof the positive

Integer nis larger than $\sqrt[\mathbf{3}]{\mathbf{n}}$, then$\raisebox{1ex}{$\mathbf{n}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{p}$}\right.$is prime or equal to 1.

Short answer

It is proved that n/pmust be a prime or 1

Step-by-step solution

Prime number

Prime: an integer greater than 1 with exactly two positive integer divisors 1 and the number itself.

Proving that  n/p is prime or equal to  

Given that p is the smallest prime factor of and larger than $\sqrt[3]{n}$

Therefore,

$$\begin{gathered} p>\sqrt[3]{n} \\ {p}^3>n \end{gathered}$$

According to this, a minimum of 3 products of the prime p is greater than n .

So, there can be no more than one other prime factor of n

Therefore n/pmust be a prime or 1