2019 · Q2
What is the largest value of n such that 10ⁿ divides the product 2⁵ × 3³ × 4⁸ × 5³ × 6⁷ × 7⁶ × 8¹² × 9⁹ × 10⁶ × 15¹² × 20¹⁴ × 22¹¹ × 25¹⁵?
Elementary MathematicsNumber System
- A65
- B55
- C50
- D45
Explanation
10^n dividing the product needs n factors 2 and n factors 5. Counting 5s: 5^3→3, 10^6→6, 15^12→12, 20^14→14, 25^15→30, total 65. Counting 2s: 2^5→5, 4^8→16, 6^7→7, 8^12→36, 10^6→6, 20^14→14, 22^11→11, total 95 ≥ 65. Hence the limiting factor is 5 and n = 65.