Problem 6 Show that $a b /(a, b) \mid m$... [FREE SOLUTION]

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Elementary Number Theory

Wissam Rajj

$Math Studyset 91影视 Explanations$ Math

2020 Edition

Chapter 2: Problem 6

Show that $a b /(a, b) \mid m$ where $m=$.

Short Answer

Expert verified

Question: Prove that $\frac{a b}{\operatorname{gcd}(a, b)} \mid \operatorname{lcm}(a, b)$. Answer: Since $\operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b) = a b$, it is clear that $\frac{a b}{\operatorname{gcd}(a, b)}$ must divide $\operatorname{lcm}(a, b)$.

Step by step solution

Prime Factorization

Write down the prime factorizations of $a$ and $b$: $a = p_1^{\alpha_1} \cdot p_2^{\alpha_2} \cdots p_n^{\alpha_n}$ $b = p_1^{\beta_1} \cdot p_2^{\beta_2} \cdots p_n^{\beta_n}$

Find the GCD

In order to find the gcd of $a$ and $b$, we take the minimum power of each prime factor: $\operatorname{gcd}(a, b) = p_1^{\min\{\alpha_1, \beta_1\}} \cdot p_2^{\min\{\alpha_2, \beta_2\}} \cdots p_n^{\min\{\alpha_n, \beta_n\}}$

Find the LCM

Similarly, in order to find the lcm of $a$ and $b$, we take the maximum power of each prime factor: $\operatorname{lcm}(a, b) = p_1^{\max\{\alpha_1, \beta_1\}} \cdot p_2^{\max\{\alpha_2, \beta_2\}} \cdots p_n^{\max\{\alpha_n, \beta_n\}}$

The Product

The product of the gcd and lcm can be written as follows: $\operatorname{gcd}(a, b) \cdot \operatorname{lcm}(a, b) = \Big(p_1^{\min\{\alpha_1, \beta_1\}} \cdot p_2^{\min\{\alpha_2, \beta_2\}} \cdots p_n^{\min\{\alpha_n, \beta_n\}}\Big) \times \Big(p_1^{\max\{\alpha_1, \beta_1\}} \cdot p_2^{\max\{\alpha_2, \beta_2\}} \cdots p_n^{\max\{\alpha_n, \beta_n\}}\Big)$

Show Divisibility

Combine the powers of the prime factors in the product to get: $p_1^{\alpha_1 + \beta_1} \cdot p_2^{\alpha_2 + \beta_2} \cdots p_n^{\alpha_n + \beta_n}$ Now, note that this is equal to $ab$, implying that $ab \mid \operatorname{gcd}(a, b) \operatorname{lcm}(a, b)$. Therefore, we can write $\frac{a b}{\operatorname{gcd}(a, b)} \mid \operatorname{lcm}(a, b)$, which proves the given statement.

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91影视

Show that \(a b /(a, b) \mid m\) where \(m=\).

Short Answer

Step by step solution

Prime Factorization

Find the GCD

Find the LCM

The Product

Show Divisibility

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