2000 PMWC Problems/Problem I7

Problem

$a$ and $b$ are two numbers that have prime factors $3$ and $5$ only. $a$ has $12$ factors ( $1$ and itself are included), $b$ has $10$ factors ( $1$ and itself are included), and their HCF (Highest Common Factor) is $75$ . What is the LCM (Least Common Multiple of $a$ and $b$ ?

Solution

Notice that in prime factorizations, one of $a$ or $b$ has to equal $3^15^x$ , and the other must equal $3^y5^2$ . This is due to $75=3^1\cdot5^2$ . Notice that the number equal to $3^15^x$ has $2(x+1)$ factors, and $3^y5^2$ has $3(y+1)$ factors. Since only the latter is divisible by $3$ , $3(y+1)=12$ , so $y=3$ . Thus $2(x+1)=10$ , so $x=4$ . Then our numbers are $a=3^35^2$ and $b=3^15^4$ . The LCM is thus $3^35^4=27\cdot625=\boxed{16875}$ . ~ eevee9406

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2000 PMWC Problems/Problem I7

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