The highest common factor of x and y is 24 and lowest common multiple of x and y is a multiple of 15. Given that x>y, x>24 and y>24, find the values of x and y

We are given the following information about two numbers \(x\) and \(y\):

- The highest common factor (HCF) of \(x\) and \(y\) is 24.
- The lowest common multiple (LCM) of \(x\) and \(y\) is a multiple of 15.
- \(x > y\), \(x > 24\), and \(y > 24\).

### Step 1: Express \(x\) and \(y\) in terms of their HCF
Since the HCF of \(x\) and \(y\) is 24, we can express \(x\) and \(y\) as:
\[
x = 24a \quad \text{and} \quad y = 24b
\]
where \(a\) and \(b\) are co-prime integers (i.e., the HCF of \(a\) and \(b\) is 1). This is because if 24 is the HCF, it means that 24 divides both \(x\) and \(y\), and the remaining factors of \(x\) and \(y\) (which are \(a\) and \(b\)) must be co-prime.

### Step 2: Analyze the LCM condition
The LCM of \(x\) and \(y\) is given by:
\[
\text{LCM}(x, y) = \frac{x \times y}{\text{HCF}(x, y)} = \frac{24a \times 24b}{24} = 24 \times a \times b
\]
We are told that this LCM is a multiple of 15, meaning:
\[
24ab = 15k
\]
for some integer \(k\). Thus, we need \(ab\) to contain the factors of 15, which means \(ab\) must be divisible by both 3 and 5.

### Step 3: Determine possible values for \(a\) and \(b\)
Since \(a\) and \(b\) are co-prime, and \(ab\) must be divisible by both 3 and 5, one of \(a\) or \(b\) must be divisible by 3, and the other must be divisible by 5. 

The simplest co-prime pairs that satisfy this condition are:
\[
a = 3, \, b = 5 \quad \text{or} \quad a = 5, \, b = 3
\]

### Step 4: Calculate \(x\) and \(y\)
Using these values of \(a\) and \(b\), we can now calculate \(x\) and \(y\).

1. If \(a = 3\) and \(b = 5\), then:
   \[
   x = 24a = 24 \times 3 = 72 \quad \text{and} \quad y = 24b = 24 \times 5 = 120
   \]
   
2. If \(a = 5\) and \(b = 3\), then:
   \[
   x = 24a = 24 \times 5 = 120 \quad \text{and} \quad y = 24b = 24 \times 3 = 72
   \]

### Step 5: Conclusion
Since \(x > y\), the correct pair is:
\[
x = 120 \quad \text{and} \quad y = 72
\]

Thus, the values of \(x\) and \(y\) are \(x = 120\) and \(y = 72\).

Would you like more details on any step, or have any questions?

Here are 5 related questions for further exploration:
1. How do you find the highest common factor (HCF) of two numbers?
2. How can you use prime factorization to find the LCM of two numbers?
3. What are co-prime numbers, and why are they important in this problem?
4. Can the LCM of two numbers ever be smaller than either of the numbers?
5. How does the relationship between HCF and LCM help solve problems involving pairs of numbers?

**Tip:** When solving problems involving HCF and LCM, expressing the numbers in terms of their HCF can greatly simplify finding solutions.

The highest common factor of x and y is 24 and the lowest common multiple of x and y is a multiple of 15. Given that x > y, x > 24, and y > 24, find the values of x and y.

Learn how to find the values of x and y when their greatest common divisor (GCD) is 24, and the least common multiple (LCM) is a multiple of 15. This problem explores key concepts such as GCD, LCM, and coprime numbers, with a detailed step-by-step solution. Suitable for Grades 8-10.

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