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How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000,

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Joined: 19 Oct 2018
Posts: 1080
Location: India
How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000,  [#permalink]

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15 Oct 2019, 16:20
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How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000, LCM(b, c) = 2000, LCM (c,a) = 2000
and HCF (a, b) = k × 125, where k is a positive integer?

A. 70
B. 40
C. 32
D. 28
E. 16
Math Expert
Joined: 02 Aug 2009
Posts: 8201
Re: How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000,  [#permalink]

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15 Oct 2019, 18:49
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nick1816 wrote:
How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000, LCM(b, c) = 2000, LCM (c,a) = 2000
and HCF (a, b) = k × 125, where k is a positive integer?

A. 70
B. 40
C. 32
D. 28
E. 16

Since HCF(a,b) $$= 125*k=5^3*k$$ and LCM(a,b)=$$1000=5^3*2^3$$, at least one of a and b is a multiple of 2^3..
Also as all LCMs are 1000 and 2000, this means a, b and c are multiples of only 2 and 5..

(I) If a is $$2^3$$, b can be $$2^0,2^1,2^2,2^3$$---1*4=4ways
(II) If b is $$2^0,2^1,2^2$$, b will be $$2^3$$---3*1=3ways
So total ways of (a,b)=4+3=7ways

Value of c for each of the 7 pairs of (a,b)
As LCM when c is included is 2000 or $$2^45^3$$, while that of (a,b) is 1000, c is surely multiple of $$2^4$$ and it can have any power of c as $$5^3$$ is already included in a and b, so c can be $$2^45^0,2^45^1,2^45^2,2^45^3$$---4 ways
So for each pair of (a,b), c can have 4 ways..Total = 7*4=28 ways

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Re: How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000,   [#permalink] 15 Oct 2019, 18:49
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How many ordered triplets (a, b, c) exist such that LCM (a, b) = 1000,

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