(Solved): 5.6 please Definition. The prime power factorization of a natural number n is found by grouping the ...

Definition. The prime power factorization of a natural number $n$ is found by grouping the common primes together in the prime factorization. For example, $12101040=2^4\cdot 3^2\cdot 5\cdot 7^5$. In general, we might write $n=p_1^{a_1}{p}_2^{a_2}\cdots p_k^{a_k}$, where it is understood that $p_1,p_2,\dots ,p_k$ are distinct primes. This method of writing a generic prime power factorization of an integer will be useful for proving the theorems below. In particular, if we are comparing the prime factorizations of two integers $a$ and $b$, it is useful to let $p_1,p_2,\dots ,p_k$ be the set of all primes which divide either $a$ or $b$, and then write $a=p_1^{a_1}{p}_2^{a_2}\cdots p_k^{a_k}$ and $b=p_1^{b_1}{p}_2^{b_2}\cdots p_k^{b_4}$ where each integer exponent $a_i,b_i\ge 0$. Theorem 5.5. Let $a$ be a natural number and let $p$ be a prime. If $p^4\mid a^3$, then $p^2\mid a$. Theorem 5.6. If $a$ and $b$ are natural numbers with $a^2\mid b^2$, then $a\mid b$. Theorem 5.7. Let $a,b$ and $c$ be natural numbers. If $a^2=bc$ and $\gcd (b,c)=1$, then there exist natural numbers $s$ and $t$ such that $b=s^2$ and $c=t^2$. Theorem 5.8. The number $\sqrt[3]{5}$ is irrational. (Hint: consider a proof by contradiction.)