Find the number of ordered triplets (a, b, c)
of positive integers for which LCM (a, b)
=1000, LCM (b, c) = 2000 and LCM (c, a) =
2000.
60
40
80
50
70

• 1000 = 2^3 * 5^3
  2000 = 2^4 * 5^3
  Since LCM of a and b is 1000, they at the most contain 2 three times and 5 three times.
  Also the LCM of terms containing C is 2000, the term c should contain 2 four times, as a and b can contain 2 only three times.
  so c should either be
  2^4 = 16,
  2^4 * 5^1 = 80
  2^4 * 5^2 = 400
  2^4 * 5^3 = 2000.
  
  Either a or b should have 2^3. Hence fixing 1 as 2^3 other can take values from 2^0-2^3.
  Hence total combination= 2*4 - 1(the one combination is 2^3,2^3 which we took twice)= 7
  
  2 of the three should have power of 5 as 5^3, whereas the third can take 4 values in terms of powers of 5(5^0 - 5^3). Hence total no of such arrangement = 3!/2! *4 -2(this is for 5^3,5^3,5^3 which we took thrice) = 12 -2 =10
  
  Hence ordered triplets will be 10*7 = 70
• 12 years agoHelpfull: Yes(7) No(3)
• 
  I think 70 ordered triplets are possible.
  
• 12 years agoHelpfull: Yes(3) No(1)
• 50 ordered triplets are possible.
• 12 years agoHelpfull: Yes(1) No(3)
• how..plz exlain
• 12 years agoHelpfull: Yes(0) No(3)
• pls xplain......
• 12 years agoHelpfull: Yes(0) No(0)
• plz ... explain it
• 12 years agoHelpfull: Yes(0) No(0)
• Lets first Factorize of 1000.
  
  1000 = 2^3 * 5^3
  2000 = 2^4 * 5^3
  
  Since LCM of A and B is 1000, they at the most contain 2 three times and 5 three times.
  
  Also the LCM of terms containing C is 2000, the term C should contain 2 four times, as A and B can contain 2 only three times.
  so C should either be
  2^4 = 16,
  2^4 * 5^1 = 80
  2^4 * 5^2 = 400
  2^4 * 5^3 = 2000. This option is not possible as A and B should be within 1000(their LCM is 1000)
  
  Case1: When C = 16,
  Either of A or B should have a 5^3. So A or B can be
  5^3 = 125
  5^3 * 2^1 = 250
  5^3 * 2^2 = 500
  5^3 * 2^3 = 1000
  
  Case1: When C = 80,
  Either of A or B should have a 5^3. So A or B can be
  5^3 = 125
  5^3 * 2^1 = 250
  5^3 * 2^2 = 500
  5^3 * 2^3 = 1000
  
  Case1: When C = 400,
  Either of A or B should have a 5^3. So A or B can be
  5^3 = 125
  5^3 * 2^1 = 250
  5^3 * 2^2 = 500
  5^3 * 2^3 = 1000
  
  Now Let one term say A be = 125, then B can only be 1000. LCM(A,B) = 1000
  If A = 250, B = 1000
  A = 500, B = 1000
  A = 1000, B = 1000
  This means that either of the terms A or b should be 1000. The other term can be one of 125, 250, 500 or 1000. C can either be 16, 80 or 400.
  
  So number of combinations =
  If A = 1000, 1*4*3 = 12 (A can take 1 value, B can take 4 and C can take 3)
  If B = 1000, 3*1*3 = 9 ( A can take 3 values (excluding 1000, as it was already included in the previous combination), B 1 and C 3)
  Total Number of triplets = 12 + 9 = 21
  
  (1000, 125, 16), (1000, 250, 16), (1000, 500, 16), (1000, 1000, 16)
  (1000, 125, 80), (1000, 250, 80), (1000, 500, 80), (1000, 1000, 80)
  (1000, 125, 400), (1000, 250, 400), (1000, 500, 400), (1000, 1000, 400)
  
  (125, 1000, 16), (250, 1000, 16), (500, 1000, 16)
  (125, 1000, 80), (250, 1000, 80), (500, 1000, 80)
  (125, 1000, 400), (250, 1000, 400), (500, 1000, 400)
• 12 years agoHelpfull: Yes(0) No(7)