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Question Number 209719 by hardmath last updated on 19/Jul/24

$$\mathrm{If}:$$$$7^{243}=\overline{\mathrm{a}...\mathrm{bc}}$$$$\mathrm{Find}:$$$$\mathrm{b}\cdot \mathrm{c}=?$$

Answered by MM42 last updated on 19/Jul/24

$$7^4\stackrel{100}{\equiv }1\Rightarrow 7^{240}\stackrel{100}{\equiv }1\Rightarrow 7^{243}\stackrel{100}{\equiv }{7}^3\stackrel{100}{\equiv }43$$$$\Rightarrow bc=12✓$$$$$$

Commented by hardmath last updated on 19/Jul/24

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professor},\mathrm{but}\mathbf{a}=?$$

Commented by MM42 last updated on 19/Jul/24

$$Idontknow$$

Commented by Frix last updated on 22/Jul/24

$${\log }_{10}{7}^{243}={243\log }_{10}7\approx 243\times .845098\approx$$$$\approx 205.359$$$$7^{243}\approx {10}^{205}\times {10}^{.359}$$$${10}^{.359}\approx 2.29$$$$7^{243}\approx 2.29\times {10}^{205}$$$$a=2$$

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