The HCF and LCM of the three numbers are 28 and 560 respectively. Given that two of the numbers are 112 and 140, find the third number
To find the third number, we can use the relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of three numbers. Let's call the three numbers A, B, and C, with A = 112 and B = 140, and we need to find C.
The relationship between the HCF and LCM of three numbers is given by the formula:
HCF(A, B, C) * LCM(A, B, C) = A * B * C
We are given that HCF(A, B, C) = 28 and LCM(A, B, C) = 560.
Since we know A and B, we can rewrite the equation in terms of C:
28 * 560 = 112 * 140 * C
Now, let's calculate the product of 28 and 560:
28 * 560 = 15680
So now we have:
15680 = 112 * 140 * C
We can simplify the equation further by dividing both sides by the product of 112 and 140:
15680 = 112 * 140 * C
15680 / (112 * 140) = C
Let's calculate the denominator:
112 * 140 = 15680
So by dividing the original equation by 15680, we get:
C = 15680 / 15680
C = 1
This indicates that the third number must actually be the HCF itself since multiplying it by 1 gives us the expected LCM when combined with the other two numbers. However, this seems unlikely because it would not be a unique third number, and typically, HCF refers to the greatest common factor among ALL numbers not equal to the numbers themselves unless they are all multiples of the HCF.
Let's calculate the product of A and B and divide it by the HCF to find the correct value of C:
C = (A * B) / HCF(A, B, C)
Substitute the given values into the formula:
C = (112 * 140) / 28
First, simplify by canceling out the 28 with 112:
112 / 28 = 4
4 * 140 = 560
Now divide the obtained product by 28:
C = 560 / 28
C = 20
So the third number C should be 20.
However, let's validate that the LCM of 112, 140, and 20 is 560, as given:
1. The prime factors of 112 are 2^4 * 7.
2. The prime factors of 140 are 2^2 * 7 * 5.
3. The prime factors of 20 are 2^2 * 5.
The LCM is calculated by taking the highest powers of all prime factors present in any of the numbers:
LCM = 2^4 * 7 * 5 = 16 * 7 * 5 = 560
The calculated LCM of 560 matches the given LCM, which confirms that the third number is indeed 20.