Exercise 1.2.27 (Solutions of $(a,b) = 10$ and $[a,b] = 100$ )

Question

Find positive integers $a$ and $b$ satisfying the equations $(a,b) = 10$ and $[a,b] = 100$ simultaneously. Find all solutions.

Richard Ganaye · Accepted Answer

\begin{proof} Suppose that $a \wedge b = 10$ and $a \vee b = 100$. From $a \wedge b = 10$, we deduce that there exist integers $A, B$ such that $a = 10A, \ b = 10 B$ and $A \wedge B = 1$. Then, using $A \wedge B = 1$, by Theorem 1.13,
$$100 = a \vee b = 10 A \vee 10 B = 10(A \vee B) = 10 AB.$$
Therefore $AB = 10$, where $A \wedge B = 1$, thus $A \mid 10$, $A \in \{1,2,5,10\}$ and $B = 10/A$, so
$$(A,B) \in \{(1,10),(2,5),(5,2),(10,1)\},$$
and, since $a = 10 A, b = 10 B$, 
$$(a,b) \in \{(10,100), (20, 50), (50, 20),  (100, 10)\}.$$
Conversely, these four ordered pairs are solutions of $a \wedge b = 10,\ a \vee b = 100$.
\end{proof}

Exercise 1.2.27 (Solutions of $(a,b) = 10$ and $[a,b] = 100$ )

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Exercise 1.2.27 (Solutions of $(a,b) = 10$ and $[a,b] = 100$ )

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