The discussion at How many integers are neither multiples of 2 nor... suggests that all common multiples of a and b are multiples of $lcm(a,b)$ and this seems to be true. Certainly all multiples of $lcm(a,b)$ are multiples of a and b, but how do I show that there are no others?
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I don't understand... You are asking why every common multiple of $a$ and $b$ is necessarily a multiple of the least common multiple of $a,b$? – JMoravitz Aug 28 '20 at 02:26
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@JMoravitz, that seems to be accurate. The question in the title is certainly both misleading and nonsensical. – Dylan C. Beck Aug 28 '20 at 02:27
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1@JMoravitz It's like asking whether every common factor of $a$ and $b$ is a factor of $\gcd(a,b)$; true but not entirely trivial. – Angina Seng Aug 28 '20 at 02:27
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If you meant to ask something different then please clarify and we can reopen it then. – Bill Dubuque Aug 28 '20 at 02:36
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@Carlo, understand the question as "how many natural numbers less than N are multiples of both a and b". It is obvious that you have to limit the numbers to be checked. – Anna Naden Aug 28 '20 at 02:41
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Let $m=ac$ be a multiple of $a$. Then $m$ is also a multiple of $b$ iff $$ac\equiv0\pmod b.$$ Solving for $c$, this is equivalent to $$c\equiv0\pmod{\frac{b}{\gcd(a,b)}}.$$ So the admissible $c$ are precisely the multiples of $b/\gcd(a,b)$, and so the common multiples $m$ of $a$ and $b$ are precisely the multiples of $ab/\gcd(a,b)$, that is the multiples of one fixed value, the least common multiple of $a$ and $b$.
Angina Seng
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