Question

If two positive integers p and q are written as p=a²b³ and
q=a³b² and a and b are two prime numbers, then verift that LCM(a,b)*
HCF(a,b)=pq​

Answer 1

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☆ Given:-

• p = a²b³ ---(1)

• q = a³b² ---(2)

☆ To verify:-

• HCF(p,q) × LCM(p,q) = pq.

☆ Proof:-

▪︎ It is given that,

=》 p = a²×b³

=》 p = a×a×b×b×b.

▪︎ Similarly,

=》 q = a³b²

=》 q = a×a×a×b×b.

▪︎ Therefore,

• HCF(p,q)

= a×a×b×b

= a²×b².

• LCM(p,q)

= a×a×a×b×b×b

= a³×b³.

☆ So,

HCF(p,q)×LCM(p,q)

= a²×b²×a³×b³

=a×a×b×b×a×a×a×b×b×b.

=a×a×b×b×b×a×a×a×b×b.

= a²b³ × a³b²

= p × q

{From (1) and (2)}

${\large{\blue{\boxed{\bold{=\:pq}}}}}$

▪︎ Hence Verified:D

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