If the LCM of p and q is r^2t^4s^2,where r, s | vedclass

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(C) The $LCM$ of $p$ and $q$ is given as $r^2t^4s^2$.For a prime factor $p_i^n$ in the $LCM$,if $p = p_i^{a}$ and $q = p_i^{b}$,then $\max(a, b) = n$.

Vedclass · Accepted Answer

$225$

Vedclass · Answer

(C) The $LCM$ of $p$ and $q$ is given as $r^2t^4s^2$.For a prime factor $p_i^n$ in the $LCM$,if $p = p_i^{a}$ and $q = p_i^{b}$,then $\max(a, b) = n$. The possible pairs $(a, b)$ are $(0, n), (1, n), \dots, (n, n)$ and $(n, 0), (n, 1), \dots, (n, n-1)$.This gives a total of $(n+1) + n = 2n+1$ possible pairs for each prime factor.For $r^2$,the number of pairs is $2(2)+1 = 5$.For $t^4$,the number of pairs is $2(4)+1 = 9$.For $s^2$,the number of pairs is $2(2)+1 = 5$.Since the choices for each prime factor are independent,the total number of ordered pairs $(p, q)$ is $5 	imes 9 	imes 5 = 225$.

If the $LCM$ of $p$ and $q$ is $r^2t^4s^2$,where $r, s,$ and $t$ are prime numbers,and $p$ and $q$ are positive integers,then the number of ordered pairs $(p, q)$ is ....

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