The HCF and LCM of two numbers are 2^5
 and 2^5x7^3 respectively. If one of the 
two numbers is 2^7
, then the other number 

Let's denote the two numbers as \( a \) and \( b \). We are given the following information:

- The HCF (Highest Common Factor) of \( a \) and \( b \) is \( 2^5 \).
- The LCM (Least Common Multiple) of \( a \) and \( b \) is \( 2^5 \times 7^3 \).
- One of the numbers, say \( a \), is \( 2^7 \).

### Step-by-Step Solution:

The relationship between the HCF, LCM, and the two numbers \( a \) and \( b \) is given by:
\[
\text{HCF} \times \text{LCM} = a \times b
\]

Plugging in the given values:
\[
2^5 \times (2^5 \times 7^3) = 2^7 \times b
\]

Simplifying the left-hand side:
\[
2^{10} \times 7^3 = 2^7 \times b
\]

We can isolate \( b \) by dividing both sides by \( 2^7 \):
\[
b = \frac{2^{10} \times 7^3}{2^7}
\]

Simplify the expression:
\[
b = 2^{10-7} \times 7^3 = 2^3 \times 7^3
\]

Calculating further:
\[
b = 8 \times 343 = 2744
\]

### Conclusion:
The other number is **2744**.

Would you like a detailed explanation of any step, or have further questions?

### Follow-Up Questions:
1. How do we find the HCF and LCM of two numbers in general?
2. What is the prime factorization method for finding HCF and LCM?
3. Why is the relation \( \text{HCF} \times \text{LCM} = a \times b \) always true?
4. How can we verify if two numbers are coprime?
5. Can the HCF of two numbers be greater than one of the numbers?

### Tip:
Remember, the HCF is the highest power of common prime factors, while the LCM includes the highest powers of all prime factors present in either number.

The HCF and LCM of two numbers are 2^5 and 2^5x7^3 respectively. If one of the two numbers is 2^7, then the other number is?

Learn how to find the other number when the HCF and LCM of two numbers are given, and one of the numbers is 2^7. This step-by-step solution covers key concepts like prime factorization, HCF and LCM relationships, and number theory principles. Suitable for students in Grades 9-12.

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