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February 22, 2026
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Calculate \( \ln(24) \):
\[ \ln(24) \approx 3.1781 \]
\[ r = \frac{3.1781}{10} \approx 0.3178 \]
\[ r \approx 31.78\% \]
#### 0.3178
An immunologist is studying the immune response to cancer and observes that a particular treatment doubles the number of active T-cells every 5 days. If the initial count of active T-cells is 1,000, how many active T-cells will there be after 20 days? Use the formula for exponential growth: \( N(t) = N_0 \times 2^{t/T} \), where \( T \) is the doubling time. Calculate \( N(t) \).
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Try small values modulo 5 first:
Mod 5: \(n^3 \equiv 13 \equiv 3 \pmod{5}\)
Check cubes mod 5:
\(0^3 = 0\)
\(1^3 = 1\)
\(2^3 = 8 \equiv 3\)
\(3^3 = 27 \equiv 2\)
\(4^3 = 64 \equiv 4\)
So only \(n \equiv 2 \pmod{5}\) works.
Now lift to mod 25: Let \(n = 5k + 2\). Plug into \(n^3 \equiv 13 \pmod{25}\):
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