Problem 5 · Factor to find a remainder · 4 points
Made by AI and not reviewed by a real person.
What is the remainder when 1 + 2 + 2² + ··· + 2²⁰²⁶ is divided by 2¹⁰¹³ + 3?
- 8
- 18
- 0
- 1
- 17
Show a hint
- Look for a shorter way to describe the entire sum before doing any division.
- Let x = 2¹⁰¹³. What could you subtract from x² to make x + 3 a factor?
Show the answer and solution
Answer: E — 17
The sum is 2²⁰²⁷ − 1 = 2x² − 1, where x = 2¹⁰¹³. Since x² − 9 = (x − 3)(x + 3), we get 2x² − 1 = 2(x − 3)(x + 3) + 17. Because 17 is smaller than x + 3, the remainder is 17. The tempting answer 8 comes from using x² − 9 but forgetting the factor of 2.