Finding the Possible Values of x for LCM(10, 15, x) 90

Introduction to Least Common Multiples (LCM)

Least Common Multiple (LCM) is an essential concept in mathematics often used in various fields, including algebra, number theory, and engineering. It represents the smallest positive integer that is divisible by a set of numbers without leaving a remainder. In this article, we'll explore how to determine the possible values of x so that LCM(10, 15, x) equals 90.

Prime Factorization and LCM Calculation

To find the value of x, we first need to determine the prime factorization of the given numbers. Let's break down the numbers 10, 15, and 90 into their prime factors: 10 2 x 5 15 3 x 5 90 2 x 32 x 5 Based on these prime factorizations, we can calculate the LCM of 10 and 15:

LCM(10, 15) 21 x 31 x 51 30

Now, we need the LCM of 30 and x to be 90. The LCM is determined by taking the highest power of each prime factor present in the numbers.

Conditions for the Highest Power of Each Prime Factor

For the LCM of 30 and x to equal 90, the following conditions must be met: The highest power of 2 must be 21 The highest power of 3 must be 32 The highest power of 5 must be 51

Analysis of Each Prime Factor

Let's analyze the conditions for each prime factor:

Factor 2

The LCM of 30 (which has 21) and x must have 21. Therefore, x can have either 20 or 21. It can’t have 22 or higher since it would increase the LCM beyond 90.

Factor 3

The LCM of 30 (which has 31) and x must have 32. Therefore, x must include 32.

Factor 5

The LCM of 30 (which has 51) and x must have 51. Therefore, x can have either 50 or 51.

Possible Forms of x

Given the above conditions, x must be of the form:

x 32 x 2a x 5b

Where:
a can be 0 or 1 for the factor of 2
b can be 0 or 1 for the factor of 5

Thus, the possible values for x are:

If a 0 and b 0: x 32 x 20 x 50 9 If a 0 and b 1: x 32 x 20 x 51 9 x 5 45 If a 1 and b 0: x 32 x 21 x 50 9 x 2 18 If a 1 and b 1: x 32 x 21 x 51 9 x 2 x 5 90

Summary of Possible Values for x

Thus, the possible values of x are 9, 18, 45, and 90.

Conclusion

To maintain the LCM of 10, 15, and x as 90, x can be any of these values. By understanding the prime factorization and the conditions for each prime factor, we can accurately determine the possible values of x.

Frequently Asked Questions (FAQs)

Q: Can the value of x be 180?

No, 180 is not a valid value for x. Since the highest power of 2 is 21, any additional power of 2 would increase the LCM beyond 90.

Q: Is there a shortcut to find the LCM of 10, 15, and x?

No, the detailed method described here is necessary to ensure that the LCM of 30 and x is exactly 90. However, using a calculator or software can help in validating the results.

Q: What is the hcf of 30 and 90?

The HCF (Greatest Common Factor) of 30 and 90 is 30. When one of the numbers is equal to the HCF, the other number is equal to the LCM. In this case, x is 90.