Question: A paleontologist dates a fossil using a sequence of radiometric decay steps modeled by $2^n \equiv 12 \pmod{25}$. What is the smallest positive integer $n$ satisfying this congruence?

["Title: Solving a Radiometric Dating Puzzle: Smallest $n$ Such That $2^n \equiv 12 \pmod{25}$", "In the field of paleontology, accurately dating ancient fossils is crucial for understanding Earth’s biological history. Advanced dating techniques often rely on radiometric decay, which can be modeled mathematically using modular arithmetic—particularly exponential congruences. One such compelling problem involves solving the congruence:", "$$\n2^n \equiv 12 \pmod{25}\n$$", "Understanding the smallest positive integer $ n $ that satisfies this equation not only strengthens mathematical reasoning but also illuminates the scientific process behind fossil dating using radioactive isotopes.", "---", "### Understanding the Problem", "We seek the smallest positive integer $ n $ such that when $ 2^n $ is divided by 25, the remainder is 12. Since 25 is not prime, solving exponential congruences requires testing powers of 2 modulo 25 until the target residue $ 12 $ appears.", "This method is analogous to how paleontologists use decay chains and half-lives—tracking discrete, predictable changes over “steps” until a measurable signal (like a given residue) is observed.", "---", "### Step 1: Compute Powers of 2 Modulo 25", "We compute successive powers of 2 modulo 25:", "- $ 2^1 = 2 \mod 25 = 2 $\n- $ 2^2 = 4 \mod 25 = 4 $\n- $ 2^3 = 8 \mod 25 = 8 $\n- $ 2^4 = 16 \mod 25 = 16 $\n- $ 2^5 = 32 \mod 25 = 7 $\n- $ 2^6 = 2 \cdot 7 = 14 \mod 25 = 14 $\n- $ 2^7 = 2 \cdot 14 = 28 \mod 25 = 3 $\n- $ 2^8 = 2 \cdot 3 = 6 \mod 25 = 6 $\n- $ 2^9 = 2 \cdot 6 = 12 \mod 25 = 12 $", "At $ n = 9 $, we find:", "$$\n2^9 \equiv 12 \pmod{25}\n$$", "---", "### Verification", "Let’s confirm:", "$$\n2^9 = 512,\quad 512 \div 25 = 20 \ ext{ remainder } 12,\ \ ext{so } 512 \equiv 12 \pmod{25}\n$$", "This confirms the solution.", "---", "### Smallest Positive Solution", "The powers of 2 modulo 25 cycle with a period dividing $ \phi(25) = 20 $, by Euler’s theorem. But we only need the smallest $ n $, which we have found to be $ n = 9 $. No smaller positive exponent yields $ 2^n \equiv 12 \mod{25} $.", "---", "### Conclusion", "Solving $ 2^n \equiv 12 \pmod{25} $ reveals that $ n = 9 $ is the minimal solution. This mirrors real-world paleontological work: through precise, step-by-step analysis—like testing successive decay states—we uncover precise timelines from fragmentary evidence. Understanding such mathematical principles empowers scientists and students alike to decode the deep past with greater confidence.", "Whether dating a 50-million-year-old fossil or modeling radioactive decay, modular arithmetic provides a powerful tool—proving that even ancient artifacts speak a language of exponents and residues.", "---", "Keywords: radiometric dating, $ 2^n \equiv 12 \pmod{25} $, paleontology, modular arithmetic, smallest positive $ n $, fossil dating, exponential congruence, discrete decay modeling.\nTags: paleontology, modular exponentiation, radiometric dating, number theory in science, $ n = 9 $, $ 2^n \mod 25 $"]