Question

if two positive integers p and q are written as p= a2b3 and q= a3b ;a,b are prime number , then verify LCM (p,q) x HCF (p,q)=pq

Answer 1

Pq=a2b3×a3b=a5b4
LCM pq=a2b
Hcf pq=a3b3
So lcm×hcf=a2b×a3b3=a5b4=pq
Hence proved.....

Answer 2

Solution:

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Given:

Two positive integers  p & q,.

p = a²b³,.

q = a³b,.
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To find:

The HCF of p & q,. and LCM

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We know that,

HCF of two integers = common factors to their least power (of the two integers),.

              FACTOR                l                   POWER
p & q =        a                        l                       2
                    b                        l                       1

=> HCF{p,q} = a²b

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LCM = factors to the highest power,

          
                  FACTOR                      POWER
p & q =              a                                  3
                           b                                  3

LCM{p,q} = a³b³
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Product of p& q

=> pq = (a²b³)(a³b)

=> $a^{2+3} b^{3+1}$

=> $a^5 b^4$ ..(i)

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LCM{p,q} x HCF{p,q} = product of the numbers,

=> (a³b³)(a²b) =  $a^5b^4$

=> $a^5b^4 = a^5b^4$

         => LHS = RHS

               Hence proved,.
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                              Hope it Helps !!

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