The sum and the difference of the LCM and HCF of two numbers are 312 and 264 respectively. What are the numbers if they sum up to 168.
Please answer.
Answers
The two numbers are 96 and 72.
Step-by-step explanation:
Given: The sum and the difference of the LCM and HCF of two numbers are 312 and 264 respectively.
Find: What are the numbers if they sum up to 168.
Solution:
Let x and y be the two numbers.
We know that x + y = 168 --------(1)
Sum of LCM and HCF = 312
Difference of LCM and HCF = 264.
LCM = (312 + 264) /2 = 288
Factors of 288 = 2^5 * 3^2
HCF = (312 - 264) /2 = 24
Factors of 24 = 2^3 * 3.
The possible number pairs are (32*3, 8*9) = (96, 72) or (32*9, 8*3) = (288, 24) based on the above factors.
As 2^5 or 2^3 and 3^2 or 3 need to be used.
From equation 1, we find that the correct pair of numbers is 96 and 72.
Given : The sum and the difference of the LCM and HCF of two numbers are 312 and 264 respectively. Sum of numbers = 168
To find : Numbers
Solution:
LCM + HCF = 312
LCM - HCF = 264 LCM ≥ HCF
Adding both
2(LCM) = 576
=> LCM = 288
HCF = 24
as HCF = 24
Lets assume two numbers are
24A & 24B where A & B are co prime integers
24A + 24B = 168
=> A + B = 7
Number 1 * Number 2 = LCM * HCF
=> 24 A * 24B = 288 * 24
=> AB = 12
A + B = 7 & AB = 12
only possible pair ( 3 , 4)
so numbers would be 24 * 3 , 24 * 4
= 72 & 96
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