Solving for the Unknown Number with HCF, LCM, and Given Numbers

When working with numbers, it's often necessary to find the unknown number given the Highest Common Factor (HCF), the Least Common Multiple (LCM), and one of the known numbers. This process involves utilizing the relationship between these three values, which can be expressed by the formula:

HCF * LCM First Number * Second Number

Step-by-Step Process to Find the Second Number

Identify the Values: Denote the numbers as follows: HCF as h LCM as l First number as a Second number as b

Rearrange the Formula: The formula can be rearranged to solve for b:

b h * l / a

Calculate: Substitute the values of HCF, LCM, and the first number into the formula to find the second number.

Example

Suppose we have the following values: HCF (h) 4 LCM (l) 48 First number (a) 16 Using the formula, we can find the second number (b):

b 4 * 48 / 16 192 / 16 12

Therefore, the second number is 12.

Additional Strategies for Finding the Second Number

How else can you determine the second number if given the LCM and HCF?

If the LCM of a and b is c and the HCF of a and b is d, then a * b d * c. Therefore, d (d / c) * b.

The second number must be a multiple of the given number. List the multiples and pick the lowest one that satisfies the LCM. For example, if the LCM of two numbers is 90 and one of the numbers is 15, the multiples of 15 are 30, 45, 60, 75, 90, 105, etc. Since 90 is the lowest multiple that satisfies the LCM, the second number is 90.

It is not always possible to determine the exact second number. The easiest example is when one number and the LCM are equal. In such a case, the other number can be any number that divides the LCM.

General Formula and Product of Numbers

The product of two numbers is equal to the product of their HCF and LCM. The formula is:

Other number (LCM * HCF) / first number

For instance, consider the numbers 45 and 80: Prime factorization of 45 32 * 5 Prime factorization of 80 24 * 5 HCF 5 (max exponent of prime factor 5) LCM 24 * 32 * 5 720

Product of the numbers 45 * 80 3600

Product of HCF and LCM 5 * 720 3600

Given one number as 80, the HCF as 5, and the LCM as 720, the other number can be found using the formula:

Second Number HCF * LCM / First Number

Second Number 5 * 720 / 80 45

Thus, the second number is 45.