A function f maps the set of natural numbers to whole numbers, such that f(xy)=f(x)f(y)+f(x)+f(y) for all x,y and f(p)=1 for every prime number p. Then, the value of f(160000) is
Question:
A function f maps the set of natural numbers to whole numbers, such that f(xy)=f(x)f(y)+f(x)+f(y) for all x,y and f(p)=1 for every prime number p. Then, the value of f(160000) is
A function f maps the set of natural numbers to whole numbers, such that f(xy)=f(x)f(y)+f(x)+f(y) for all x,y and f(p)=1 for every prime number p. Then, the value of f(160000) is
Options
Answer: 4095
Explanation:
Rewrite: f(xy) = (f(x)+1)(f(y)+1) − 1 → g(x) = f(x)+1 → g(xy) = g(x)g(y) g(p) = f(p)+1 = 2 Factor 160000 = 2⁸ × 5⁴ Compute g(160000) = g(2⁸) × g(5⁴) = 2⁸ × 2⁴ = 2¹² = 4096 f(160000) = g(160000) − 1 = 4095
Explanation:
Rewrite: f(xy) = (f(x)+1)(f(y)+1) − 1 → g(x) = f(x)+1 → g(xy) = g(x)g(y) g(p) = f(p)+1 = 2 Factor 160000 = 2⁸ × 5⁴ Compute g(160000) = g(2⁸) × g(5⁴) = 2⁸ × 2⁴ = 2¹² = 4096 f(160000) = g(160000) − 1 = 4095
👉 Want AI explanation? Open in MCQ App
Recommended articles
- How To Cover Current Affairs For Upsc Prelims — Master UPSC Current Affairs with daily newspaper reading, Vision IAS monthly compilations, PT 365, and MCQ practice. Complete strategy guide for Prelims…
- Top 10 Daily Current Affairs Resources For Upsc Aspirants — Discover the best daily current affairs resources for UPSC 2026. Complete guide with The Hindu, PIB, Sansad TV, and 7 more proven sources. Expert…
- Free Cat Previous Year Questions Guide — Master CAT 2025 with free previous year questions. Practice 1000+ PYQs, mock tests, and detailed solutions. Complete guide to CAT preparation using past…