Question

21.1.2 Use unique prime factorization to show that a positive integer $n$ is a square if and only if each prime in the prime factorization of $n$ occurs to an even power. 21.1.3 If $l$ and $m$ are positive integers with no common prime divisor, and $lm$ is a square, use Exercise 21.1.2 to show that $l$ and $m$ are both squares. 21.1.4 Show similarly that if $l$ and $m$ are integers with no common prime divisor, and if $lm$ is a cube, then $l$ and $m$ are both cubes.

          21.1.2 Use unique prime factorization to show that a positive integer $n$ is a square
if and only if each prime in the prime factorization of $n$ occurs to an even
power.
21.1.3 If $l$ and $m$ are positive integers with no common prime divisor, and $lm$ is a
square, use Exercise 21.1.2 to show that $l$ and $m$ are both squares.
21.1.4 Show similarly that if $l$ and $m$ are integers with no common prime divisor,
and if $lm$ is a cube, then $l$ and $m$ are both cubes.

21.1.2 Use unique prime factorization to show that a positive integer n is a square
if and only if each prime in the prime factorization of n occurs to an even
power.
21.1.3 If l and m are positive integers with no common prime divisor, and lm is a
square, use Exercise 21.1.2 to show that l and m are both squares.
21.1.4 Show similarly that if l and m are integers with no common prime divisor,
and if lm is a cube, then l and m are both cubes.

Added by Rafael D.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Chandra Jain

03/30/2022

Step 1

Suppose that $n$ is a square. Then $n = k^2$ for some integer $k$. Let $p$ be a prime that divides $n$. Then $p$ divides $k^2$, so $p$ divides $k$. Therefore, the power of $p$ in the prime factorization of $n$ is twice the power of $p$ in the prime Show more…

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21.1.2 Use unique prime factorization to show that a positive integer $n$ is a square if and only if each prime in the prime factorization of $n$ occurs to an even power. 21.1.3 If $l$ and $m$ are positive integers with no common prime divisor, and $lm$ is a square, use Exercise 21.1.2 to show that $l$ and $m$ are both squares. 21.1.4 Show similarly that if $l$ and $m$ are integers with no common prime divisor, and if $lm$ is a cube, then $l$ and $m$ are both cubes.