prove that either 2 x 100^500 +15 or 2 x 100^500 +16 is not a perfect square

To prove that either \(2 \times 100^{500} + 15\) or \(2 \times 100^{500} + 16\) is not a perfect square, let's analyze the expression and its properties modulo 4.

1. **Expressing \(100^{500}\) in modular terms:**

   Since \(100 \equiv 0 \pmod{4}\), we have
   \[
   100^{500} \equiv 0^{500} \equiv 0 \pmod{4}.
   \]
   Therefore,
   \[
   2 \times 100^{500} \equiv 2 \times 0 \equiv 0 \pmod{4}.
   \]

2. **Calculating \(2 \times 100^{500} + 15\) and \(2 \times 100^{500} + 16\) modulo 4:**

   - For \(2 \times 100^{500} + 15\):
     \[
     2 \times 100^{500} + 15 \equiv 0 + 15 \equiv 3 \pmod{4}.
     \]

   - For \(2 \times 100^{500} + 16\):
     \[
     2 \times 100^{500} + 16 \equiv 0 + 16 \equiv 0 \pmod{4}.
     \]

3. **Analyzing the possibility of each expression being a perfect square modulo 4:**

   - A perfect square modulo 4 can only be \(0\) or \(1\). This is because:
     - If \(x \equiv 0 \pmod{4}\), then \(x^2 \equiv 0 \pmod{4}\).
     - If \(x \equiv 1 \pmod{4}\), then \(x^2 \equiv 1 \pmod{4}\).
     - If \(x \equiv 2 \pmod{4}\), then \(x^2 \equiv 0 \pmod{4}\).
     - If \(x \equiv 3 \pmod{4}\), then \(x^2 \equiv 1 \pmod{4}\).

   Therefore, a perfect square cannot be congruent to \(3 \pmod{4}\).

4. **Conclusion:**

   Since \(2 \times 100^{500} + 15 \equiv 3 \pmod{4}\), it cannot be a perfect square. Hence, we conclude that **either \(2 \times 100^{500} + 15\) or \(2 \times 100^{500} + 16\) is not a perfect square** (specifically, \(2 \times 100^{500} + 15\) is not a perfect square).

Would you like more details on modular arithmetic or examples of similar problems?

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1. What are other cases where a number modulo 4 can determine if it's a perfect square?
2. How would the result change if we considered different powers, like \(100^{300}\)?
3. What is the general method for proving a number isn’t a perfect square?
4. How can properties of modular arithmetic help in solving other perfect square problems?
5. How would this proof adapt if we replaced 100 with another even number?

**Tip:** Modular arithmetic is a powerful tool for testing properties of integers, especially in proofs involving divisibility and squares.

Discover a step-by-step proof showing that either 2 x 100^500 + 15 or 2 x 100^500 + 16 cannot be a perfect square, using modular arithmetic and number theory concepts. Learn about perfect square congruences modulo 4 in this advanced high school math problem.

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