The HCF of two numbers is 15 and
their LCM is 180. If their sum is 105,
then the numbers are?
Answers
Answer:
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Step-by-step explanation:
The least of the numbers must not be less than 15
These two numbers are all multiples of 180.
We can list down multiples of 180 from 15 that are divisible by 15.
We have :
1) 15
2) 30
3) 45
4) 60
5) 90
The two possible sums are :
90 + 15 = 105
60 + 45 = 105
We have two possible pairs
90 - 15 = 75
60 - 45 = 15
The difference can either be 75 or 15
The numbers are 60 and 45.
Given data:
The HCF of two numbers is 15 and their LCM is 180. Their sum is 105.
To find:
The two numbers
Step-by-step explanation:
Since the HCF of the two numbers are 15, let the two numbers be 15x and 15y, where x, y are co-primes.
Given, LCM of 15x and 15y is 180
⇒ 15 * xy = 180
⇒ xy = 12 ... ... (1)
Also given, their sum is 105
⇒ 15x + 15y = 105
⇒ 15 (x + y) = 105
⇒ x + y = 7 ... ... (2)
We know that,
(x - y)² = (x + y)² - 4xy
⇒ (x - y)² = 7² - 4 * 12, by (1) and (2)
⇒ (x - y)² = 49 - 48
⇒ (x - y)² = 1
⇒ x - y = 1 ... ... (3)
Now, (2) + (3) ⇒
x + y + x - y = 7 + 1
⇒ 2x = 8
⇒ x = 4
Putting x = 4 in (2), we get
4 + y = 7
⇒ y = 3
So, the two numbers are (15 * 4) = 60 and (15 * 3) = 45.
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