number of ordered pairs (X,Y,Z) for positive integers satisfying LCM (X, Y) = 72 LCM (X,Z) =600 and LCM (Y,Z) = 900 is
Answers
Answer:
ordered pairs (X,Y,Z) = (24,36,300)
Step-by-step explanation:
It is given that,
number of ordered pairs (X,Y,Z) for positive integers satisfying
1) LCM (X, Y) = 72
2) LCM (X,Z) =600 and
3) LCM (Y,Z) = 900
To find x
from 1) and 2) we get x is a factor of 72 and 600
Find the HCF (72,600)
HCF (72,600) = 24
Therefore x = 24
To find y
from 1) and 3) we get y is a factor of 72 and 900
Find the HCF (72,900)
HCF (72,900) = 36
Therefore y = 36
To find z
from 2) and 3) we get z is a factor of 600 and 900
Find the HCF (600,900)
HCF (600,900) = 300
Therefore z = 300
Ordered pair (X,Y,Z) = (24,36,300)
Answer:
3
Step-by-step explanation:
It is given that,
1) LCM (X, Y) = 72
2) LCM (X,Z) =600 and
3) LCM (Y,Z) = 900
To find x
from 1) and 2) we get x is a factor of 72 and 600
Find the HCF (72,600)
HCF (72,600) = 24
Therefore x is a factor of 24 = 2³*3
To find y
from 1) and 3) we get y is a factor of 72 and 900
Find the HCF (72,900)
HCF (72,900) = 36
Therefore y is a factor of 36=2²*3²
To find z
from 2) and 3) we get z is a factor of 600 and 900
Find the HCF (600,900)
HCF (600,900) = 300
Therefore z is a factor of 300 = 2²*3*5²
Ordered pair (X,Y,Z) = (24,36,300).
On furthur analysis,
we can conclude that X should be a factor of 2³( 8 or 24)
Y should be a factor of 3²(9 or 18 or 36).
Z should be a factor of 3*5²(75, 150 or 300).
Also, X cannot take 8, since L.C.M (X,Z) will fail, hence X = 24.
Y cannot take 9 or 18 since L.C.M of (Y,Z) will fail, hence Y =36
The possible triplets are (24,36,75) ,(24,36,150) r (24,36, 150).