The least common multiple (LCM) of two positive integers is 216. If their ratio is 2:9, find their highest common factor ( HCF)
Answers
Answer:
LCM of two numbers is 216 .
Ratio of the two numbers = 2 : 9 .
Let the numbers be a and b .
Then the numbers can be written as 2 x : 9 x as they are in the ratio of 2 : 9 .
So we can write a = 2 x and b = 9 x .
We know that the product of the numbers = LCM × HCF .
HCF between 2 x and 9 x will be x because the HCF between 2 and 9 is 1 .
So the product of the numbers = 2 x × 9 x = 18 x² .
According to the question we get :
18 x² = 216 × x
⇒ 18 x² = 216 x
Cancel x both sides and we get :
⇒ 18 x = 216 .
⇒ x = 216/18
⇒ x = 12
The HCF will be 12 .
Verification :
The numbers are 2 x and 9 x .
∴ a = 24 , b = 108 .
LCM of a and b = 216
24 = 12 × 2
108 = 12 × 9
LCM = 12 × 2 × 9 ⇒ 12 × 18 ⇒ 216
HCF = 12 because it is the highest common factor between 108 and 24 .
Hence the answer is verified !