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27 Jan 2022, 07:00
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Be sure to select an answer first to save it in the Error Log before revealing the correct answer (OA)!
Difficulty:
85%
(hard)
Question Stats:
48% (02:14) correct
52%
(02:27)
wrong
based on 81
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What is the remainder when \(19^{561}\) is divided by 29?
(A) 19
(B) 18
(C) 17
(D) 9
(E) 8
Are You Up For the Challenge: 700 Level Questions
(A) 19
(B) 18
(C) 17
(D) 9
(E) 8
Are You Up For the Challenge: 700 Level Questions
20 Feb 2022, 07:51
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Remainder of \(19^{561}\) / 29
=> \(19^{561}\) = \((19^2)^{280}*19 \)= \((348+13)^{280}*19\) =
= \((13^2)^{140}*19 \)= \((174 - 5)^{140}*19 \)= \((- 5)^{140}*19 \) =
= \((5^3)^{46}*5^2*19 \)= \((116+9)^{46}*5^2*19\) =
= \(9^{45}*9*5^2*19\) = \((9^3)^{15}*9*5^2*19 \) = \((725 + 4)^{15}*9*5^2*19\) =
= \(4^{15}*9*5^2*19 \)= \((58+6)^5*9*5^2*19 \)= \(6^5*9*5^2*19 \)=
= \((6*5)^2*6^2*9*6*19\) = \(7*9*6*19 = (58+5)*6*19\) =
= \((5*6)*19 \)= 19
Option A
=> \(19^{561}\) = \((19^2)^{280}*19 \)= \((348+13)^{280}*19\) =
= \((13^2)^{140}*19 \)= \((174 - 5)^{140}*19 \)= \((- 5)^{140}*19 \) =
= \((5^3)^{46}*5^2*19 \)= \((116+9)^{46}*5^2*19\) =
= \(9^{45}*9*5^2*19\) = \((9^3)^{15}*9*5^2*19 \) = \((725 + 4)^{15}*9*5^2*19\) =
= \(4^{15}*9*5^2*19 \)= \((58+6)^5*9*5^2*19 \)= \(6^5*9*5^2*19 \)=
= \((6*5)^2*6^2*9*6*19\) = \(7*9*6*19 = (58+5)*6*19\) =
= \((5*6)*19 \)= 19
Option A
27 Feb 2022, 06:49
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An easy way to work this out would be to se Fermat Little's theorem which says that -
for any prime number p and any integer a such that p does not divide a (the pair are relatively prime), p divides exactly into a^p − a.
\(19^{28}\) ≡ 1 (mod 29)
⟹ \((19^{28})^{28}\) ≡ \(1^{28}\) (mod 29)
⟹ \(19^{560}\) ≡ 1 (mod 29)
⟹ \(19^{560}\) * 19 ≡ 19 (mod 29)
Since 19 is less than 29, the remainder is the same number and the answer is 19(A)
for any prime number p and any integer a such that p does not divide a (the pair are relatively prime), p divides exactly into a^p − a.
\(19^{28}\) ≡ 1 (mod 29)
⟹ \((19^{28})^{28}\) ≡ \(1^{28}\) (mod 29)
⟹ \(19^{560}\) ≡ 1 (mod 29)
⟹ \(19^{560}\) * 19 ≡ 19 (mod 29)
Since 19 is less than 29, the remainder is the same number and the answer is 19(A)
