Subtopic: Free & Forced Vibrations of Continuous Beams
Question Statement:
If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as:
Options:
(A) sum of squares of two natural numbers
(B) sum of cubes of two natural numbers
(C) sum of square roots of two natural numbers
(D) sum of cube roots of two natural numbers
Official Answer Key:
A
Step-by-Step Solution & Explanation
### **Dedicated Step-by-Step Solution (GATE AE 2012 Q.63)**
**1. Problem Statement & Parameters**:
- **Module**: Aerospace Structures (2 Mark, MCQ)
- **Topic**: Structural Dynamics & Aeroelasticity (Free & Forced Vibrations of Continuous Beams)
- **Organizing Institute**: IIT Delhi
- **Problem Statement**:
> If a prime number on division by 4 gives a remainder of 1, then that number can be expressed as:
**2. Mathematical Derivation & Physical Analysis**:
1. **Number Theory Principle (Fermat's Christmas Theorem)**:
According to Fermat's Theorem on Sums of Two Squares, an odd prime number $p$ can be expressed as the sum of two squares of integers:
p = x^2 + y^2 \quad (x, y \in \mathbb{N})
if and only if $p \equiv 1 \pmod 4$ (i.e. $p$ leaves a remainder of 1 upon division by 4).
2. **Empirical Verification**:
- $p = 5 = 1^2 + 2^2$
- $p = 13 = 2^2 + 3^2$
- $p = 17 = 1^2 + 4^2$
- $p = 29 = 2^2 + 5^2$
3. **Conclusion**:
The prime can be expressed as the sum of squares of two natural numbers (**Option A**).
**3. Comprehensive Options Analysis**:
- **(A)** *"sum of squares of two natural numbers"*: ✅ **CORRECT**.
- **(B)** *"sum of cubes of two natural numbers"*: ❌ **INCORRECT**.
- **(C)** *"sum of square roots of two natural numbers"*: ❌ **INCORRECT**.
- **(D)** *"sum of cube roots of two natural numbers"*: ❌ **INCORRECT**.
**4. Official Answer Verification**:
- **Official Master Answer**: **A**
- **Evaluation Status**: Matches official GATE 2012 (IIT Delhi) master answer key with 100% precision.