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Question Number 22372 by Rasheed.Sindhi last updated on 16/Oct/17

Commented by Rasheed.Sindhi last updated on 16/Oct/17

Question asked by Mr. Tinkutara.  reposted for answer.

QuestionaskedbyMr.Tinkutara.repostedforanswer.

Commented by Tinkutara last updated on 16/Oct/17

It is Q. 19634. I have solution but I did  not get understand.

ItisQ.19634.IhavesolutionbutIdidnotgetunderstand.

Commented by Rasheed.Sindhi last updated on 16/Oct/17

Pl post your answer.

Plpostyouranswer.

Commented by Tinkutara last updated on 16/Oct/17

Posted it in original post.

Posteditinoriginalpost.

Answered by Rasheed.Sindhi last updated on 22/Oct/17

Formula:    lcm(p^a .q^b .r^c  , p^d .q^e .r^f )=p^g .q^h .r^i                     ⇒a,d≤g ; b,e≤h ; c,f≤i  −−−−−−−−−−−−−  72=2^3 .3^2  , 900=2^2 .3^2 .5^2 , 600=2^3 .3.5^2   Let x=2^m .3^n .5^0 ,y=2^s .3^t .5^0 ,z=2^u .3^v .5^w   ^•  lcm(x,y)        =lcm(2^m .3^n .5^0 ,2^s .3^t .5^0 )=2^3 .3^2 .5^0    ⇒m,s≤3 ; n,t≤2;max(m,s)=3,max(n,t)=2....(i)  ^• lcm(y,z)        =lcm(2^s .3^t .5^0 ,2^u .3^v .5^w )=2^2 .3^2 .5^2    ⇒s,u≤2,t,v≤2,w≤2;max(0,w)=2,max(s,u)=2...(ii)  ^• lcm(x,z)        =lcm(2^m .3^n .5^0 ,2^u .3^v .5^w )=2^3 .3.5^2    ⇒m,u≤3;n,v≤1;w≤2..............(iii)  (i) & (ii)    s≤3 ∧ s≤2⇒s≤2...................(iv)  (i) & (iv):  max(m,s)=3 ∧ s≤2⇒m=3  (ii): w≤2 ∧ max(0,w)=2⇒w=2  (i) & (iii):n≤2 ∧ n≤1⇒n≤1.........(v)  (i)&(v):max(n,t)=2 ∧ n≤1⇒t=2  (ii)&(iv):max(s,u)=2∧s≤2⇒u≤2  So now,  x=2^(m=3) .3^(n≤1) .5^0 ,y=2^(s≤2) .3^(t=2) .5^0 ,z=2^(u≤2) .3^(v≤1) .5^2   x=2^3 .3^(0,1) ,y=2^(0,1,2) .3^2 , z=2^(0,1,2) .3^(0,1) .5^2   ∵lcm(2^(0,1,2) .3^2 ,2^(0,1,2) .3^(0,1) .5^2 )=2^2 .3^2 .5^2        y       z      2^0 →2^2      1 way       2^1 →2^2      1 way       2^2 → 2^(0,1,2)  3 ways  Total ways for power of 2=5  ∵lcm(2^3 .3^(0,1) ,2^(0,1,2) .3^(0,1) .5^2 )=2^3 .3.5^2        x        z       3^0 →3^1     1 way        3^1 →3^(0,1)    2 ways  Total ways for power of 3=3  ∴  Number of required triplets 5×3=15

Formula:lcm(pa.qb.rc,pd.qe.rf)=pg.qh.ria,dg;b,eh;c,fi72=23.32,900=22.32.52,600=23.3.52Letx=2m.3n.50,y=2s.3t.50,z=2u.3v.5wlcm(x,y)=lcm(2m.3n.50,2s.3t.50)=23.32.50m,s3;n,t2;max(m,s)=3,max(n,t)=2....(i)lcm(y,z)=lcm(2s.3t.50,2u.3v.5w)=22.32.52s,u2,t,v2,w2;max(0,w)=2,max(s,u)=2...(ii)lcm(x,z)=lcm(2m.3n.50,2u.3v.5w)=23.3.52m,u3;n,v1;w2..............(iii)(i)&(ii)s3s2s2...................(iv)(i)&(iv):max(m,s)=3s2m=3(ii):w2max(0,w)=2w=2(i)&(iii):n2n1n1.........(v)(i)&(v):max(n,t)=2n1t=2(ii)&(iv):max(s,u)=2s2u2Sonow,x=2m=3.3n1.50,y=2s2.3t=2.50,z=2u2.3v1.52x=23.30,1,y=20,1,2.32,z=20,1,2.30,1.52lcm(20,1,2.32,20,1,2.30,1.52)=22.32.52yz20221way21221way2220,1,23waysTotalwaysforpowerof2=5lcm(23.30,1,20,1,2.30,1.52)=23.3.52xz30311way3130,12waysTotalwaysforpowerof3=3Numberofrequiredtriplets5×3=15

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