Problem 10 · Combine modular cycles with exponent divisibility · 5 points
Made by AI and not reviewed by a real person.
If the chart’s exponential trend continues, what is the first whole-number hour after hour 4 when the bacteria count is a perfect fourth power and its increase since hour 4 is divisible by 49?
- 7
- 46
- 67
- 25
- 68
Show a hint
- Factor the increase since hour 4, then study its divisibility by 49 separately from the fourth-power condition.
- First check when powers of 2 equal 1 modulo 7. Then use 2³ = 1 + 7 to check which of those powers equal 1 modulo 49.
Show the answer and solution
Answer: C — 67
At hour t, the count is 2^(t+1), so its increase since hour 4 is 32(2^(t−4)−1). Because 32 is coprime to 49, the increase is divisible by 49 exactly when 2^(t−4) ≡ 1 (mod 49). Powers of 2 repeat every three exponents modulo 7, so write t−4 = 3m. Then 2^(t−4) = (1+7)^m ≡ 1+7m (mod 49), which equals 1 modulo 49 exactly when 7 divides m. Thus t = 4+21j for a positive integer j. The count is a fourth power when t+1 = 5+21j is divisible by 4, or j ≡ 3 (mod 4). The smallest such j is 3, giving hour 67. Hour 25 is tempting: its increase is divisible by 49, but its count, 2^26, is not a fourth power.