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22 Aug 2021, 04:06
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C
Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
95%
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Question Stats:
47% (02:40) correct
53%
(02:41)
wrong
based on 123
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What is the remainder when 3^21 + 9^21 + 27^21 + 81^21 is divided by (3^20 + 1)?
A. 70
B. 55
C. 60
D. 62
E. 77
A. 70
B. 55
C. 60
D. 62
E. 77
22 Aug 2021, 05:06
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Darknight123
Let us get terms of \(3^{21}+9^{21}+27^{21}+81^{21}\) in terms of \(3^{20}+1\).
Now \(3^{21}=3^{21}+3-3=3^{20}*3+3-3=3(3^{20}+1)-3\)
Now \(3(3^{20}+1)-3\) divided by \(3^{20}+1\) will leave a remainder of -3.
Also, \(9^{21}=(3^{21})^2\) will leave a remainder of \((-3)^2\) or 9.
Thus the remainder is \(-3+(-3)^2+(-3)^3+(-3)^4=-3+9-27+81=60\)
C
General Discussion
11 Nov 2023, 07:00
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Asked: What is the remainder when 3^21 + 9^21 + 27^21 + 81^21 is divided by (3^20+1)?
3^21 + 9^21 + 27^21 + 81^21 = 3^21 + 3^42 + 3^63 + 3^84 = 3^21 (1 + 3^21 + 3^42 + 3^63) = 3^21 (3^21 + 1)(3^42 + 1) = {3*(3^20+1) - 3}{3*(3^20 + 1) -2}{9*(3^20+1)^2 - 18(3^20 + 1) +10}
The remainder when 3^21 + 9^21 + 27^21 + 81^21 is divided by (3^20+1) = (-3)(-2)(10) = 60
x^3 + x^2 + x + 1 = x^2(x+1) + (x+1) = (x^2 + 1)(x+1)
3^21 = 3*(3^20 + 1) - 3
3^42 = 9*(3^20 + 1)^2 + 9 -18(3^20 + 1)
IMO C
3^21 + 9^21 + 27^21 + 81^21 = 3^21 + 3^42 + 3^63 + 3^84 = 3^21 (1 + 3^21 + 3^42 + 3^63) = 3^21 (3^21 + 1)(3^42 + 1) = {3*(3^20+1) - 3}{3*(3^20 + 1) -2}{9*(3^20+1)^2 - 18(3^20 + 1) +10}
The remainder when 3^21 + 9^21 + 27^21 + 81^21 is divided by (3^20+1) = (-3)(-2)(10) = 60
x^3 + x^2 + x + 1 = x^2(x+1) + (x+1) = (x^2 + 1)(x+1)
3^21 = 3*(3^20 + 1) - 3
3^42 = 9*(3^20 + 1)^2 + 9 -18(3^20 + 1)
IMO C
