Problem 8 · Track two interacting cycles · 5 points
Made by AI and not reviewed by a real person.
What is the remainder when 1¹ + 2² + 3³ + ⋯ + 2020²⁰²⁰ is divided by 7?
- 6
- 2
- 5
- 0
- 1
Show a hint
- Moving seven places keeps the base the same modulo 7. Does it also keep the exponent at the same place in its cycle?
- Consider 42 consecutive terms. For each fixed base modulo 7, what exponent remainders occur?
Show the answer and solution
Answer: B — 2
For a base not divisible by 7, powers repeat with a period dividing 6. In any 42 consecutive terms, each base modulo 7 occurs with all six exponent remainders modulo 6. A base of 0 contributes 0; a base of 1 contributes 6; and each base a from 2 through 6 contributes 0 because 1 + a + ⋯ + a⁵ ≡ 0 (mod 7). Thus each 42-term block sums to 6 modulo 7. Since 2020 = 48·42 + 4, the remaining four terms have remainders 1, 4, 6, and 4. The total is 48·6 + 1 + 4 + 6 + 4 = 303, leaving remainder 2. The tempting answer 6 comes from treating seven terms as a full cycle: after seven places the base repeats modulo 7, but the exponent has shifted in its cycle.