Prove that an integer n>2 is a square if and only if every exponent in the prime factorization of n is even

Question

This is a question that is on my final review sheet and cannot seem to find an answer to it.

1- Prove that an integer n>2 is a square if and only if every exponent in the prime factorization of n is even.

2- Generalize the previous statement and state a necessary and sufficient condition for an integer n>2 to be the t-th power of the some integer.

3- Find with proof the prime factorization of the smallest positive integer divisible by 2 and by 3 and which simultaneously a cube and a fifth power.

I know this is 3 in one, but I have tried for hours to figure an answer to this and still am stuck on it. Thank you for your time.

Answer 1

Some ideas to get you unstuck:

Observe that $p_1^{2{a_1}}p_2^{2{a_2}} \cdots p_n^{2{a_n}} = (p_1^{{a_1}}p_2^{{a_2}} \cdots p_n^{{a_n}})^2$. This proves one direction of (1); you can determine which one, and then attempt the other direction. Note in particular the role of $2$ in the aforementioned equation; this should suffice in stating and proving your second problem.

For the third problem, hopefully your answer to the second will indicate that $2$ and $3$ (as primes) must each be raised to powers that are multiples of $3$ and $5$ (to ensure the number is a cube and fifth power); since the least common multiple of $3$ and $5$ is $15$, this means that each of the prime divisors must be raised to a power that is a multiple of $15$. The smallest such power, of course, is $15$ itself; so, we end up with $2^{15}3^{15}$ as the minimal such number.

Answer 2

1) Let number be a^2. Then a can be written as a product of primes uniquely a= (p.q.r......) therefore a^2= (p.q.r......)^2 or a^2= (p^2)(q^2).........

2) Similarly if it is a^t= (p.q.r.......)^t= (p^t)(q^t)..........

3) (2.3)^(3.5)= 6^15