777 views
8 votes
8 votes
Find the number of combinations of (a, b, c) if LCM (a, b) = 1000, LCM (b, c) = 2000, LCM (c, a) = 2000.

I am getting total 96 combinations

2 Answers

Best answer
5 votes
5 votes

LCM (a, b) = 23×53

LCM (b, c) = 24×53

LCM (c, a) = 24×53

Therefore prime factorization shows that only 2 and 5 are involved in combination of numbers.

Let a = 2p×5q

b = 2r×5s

c = 2t×5u

Calculation for powers of 2:

Above LCM's shows that 24 came from c thus t = 4.  Now either p or r will take 3 and other will take 0, 1, 2, 3.

Total options for (p,r) = (0,3) (1,3) (2,3) (3,3) (3,0) (3,1) (3,2) = 7

Calculation for powers of 5: 
Maximum power of 5 is 3.  So any two of q, s, u have maximum power 3.  and other will take 0, 1, 2, 3.

Total options for (q,s,u) = (0,3,3) (1,3,3) (2,3,3) (3,0,3) (3,1,3) (3,2,3) (3,3,0) (3,3,1) (3,3,2) (3,3,3) = 10

Total combinations = 7 x 10 = 70

selected by
1 votes
1 votes

LCM (a, b) = $2^{3}×5^{3}$, LCM (b, c) = $2^{4}×5^{3}$ and LCM (c, a) = $2^{4}×5^{3}$

Therefore prime factorization shows that only 2 and 5 are involved in combination of numbers.

Let a = $2^{P}×5^{Q}$ , b = $2^{R}×5^{S}$ and a = $2^{T}×5^{U}$. then  answer is 70. For very detailed explanation, Please refer attached image.

Related questions

0 votes
0 votes
0 answers
1
Verendra asked Jun 8, 2018
369 views
largest four digit number which divisible by 12or 16 or 18 and leaves remainder 1.but divided by 17 remainder is 0
2 votes
2 votes
2 answers
2
Manu Thakur asked Oct 26, 2017
1,863 views
Calculate number of pairs Whose LCM is 800 and HCF is 20?
0 votes
0 votes
1 answer
3
Surya Dhanraj asked Apr 17, 2017
1,841 views
How to determine whether a question is of LCM or HCF ?? It's sometime very confusing. Please help. Thank-you in advance.
1 votes
1 votes
1 answer
4
Purple asked Jan 30, 2016
1,850 views
A natural number n is such that $120≤ n ≤ 240.$ If HCF of $n$ and $240$ is $1,$ how many values of $n$ are possible?$A)24$ $B)32$ ...