Question

If r, s, t are prime numbers and p, q are the positive integers such that the LCM of p, q is
${r}^{2}{t}^{4}{s}^{2}$, then the number of ordered pair (p,q) is :-

(a) 252

(b) 254

(c) 225

(d) 224

✔️✔️ quality answer needed✔️✔️​

Answer 1

$\huge\red{\mathfrak{ANSWER!!!!}}$

{total no  of possibilities are 5.}

{The same logic works for t. }

{There are 5 ways to split up the powers of t in p and q.For s, you will have}

{ the 9 possibilitie
sνs
pνs
q041424344443424140Altogether, }

{here are 559455 ways to build pand q from their prime factors}

hope it helps:--

T!—!ANKS!!!

Answer 2

Answer:

Option(c)

Step-by-step explanation:

Given that r,s,t are prime numbers.

p,q are positive integers.

LCM(p,q) = r²t⁴s²

Now,

(i) r² ordered pairs:

(1,r)², (r²,1), (r²,r), (r,r²),(r²,r²)

= 5 ways.

(ii) t⁴ ordered pairs:

(t⁰,t⁴),(t⁴,t⁰),(t,t⁴),(t⁴,t),(t²,t⁴),(t⁴,t²),(t³,t⁴),(t⁴,t³),(t⁴,t⁴)

= 9 ways.

(iii) s² ordered pairs:

(1,s²),(s²,1),(s²,s),(s,s²),(s²,s²)

= 5 ways.

Total number of ordered pairs:

= 5 * 9 * 5

= 225 ways.

Hope it helps!