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Find the remainder when the power tower 7^(7^7) is divided by 1000.…
Question
Find the remainder when the power tower 7^(7^7) is divided by 1000. (Here 7^7 = 823543, so you must compute 7^823543 modulo 1000.)
✓ Verified answer: 343checked by our engine — not a guess
Step-by-step solution
We work modulo 1000 = 8*125.
The multiplicative order of 7 modulo 1000: since lambda(1000) = lcm(lambda(8),lambda(125)) = lcm(2,100) = 100, and one checks ord_1000(7) = 4.
Actually compute the order: 7^2=49, 7^4=2401≡401, 7^5≡807, ...
the true order of 7 mod 1000 is 20 (since 7^20≡1).
So we need the exponent 7^7 modulo 20.
Now 7^7 mod 20: 7^2=49≡9, 7^4≡9^2=81≡1 mod 20, so 7^7 = 7^4·7^2·7 ≡ 1·9·7 = 63 ≡ 3 (mod 20).
Therefore 7^(7^7) ≡ 7^3 = 343 (mod 1000).
Final answer343
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