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Q: Find the remainder of 2^ 2019 when divided by (a) 2. (b) 5. (c) 7. (d) 9

Q: Find the remainder of 2^ 2019 when divided by (a) 2. (b) 5. (c) 7. (d) 9

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Q: Find the remainder of 2^ 2019 when divided by

(a) 2. (b) 5. (c) 7. (d) 9

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c > 7 3=3 (mod 7) = 36 = 36 (mod Z) → 36 = 729 (mod 7) → 36 = 728 +1 (mod 7) => 36= 0+1 (mod 7) => 36 = 1 (mod 7) => (36) 76 = (1) 76 (mad 7) ⇒ 3456 = f (m07) 25 19 3 = 3 (mod 9) = 3 ² = 3² (mod af ⇒ 3² = 9 (mod af 30 (mod a) → (3²) 228 = 0 (mod: a) → 3+56=0 (mod 9): 1 11 ** 1.
Transcribed Image Text:c > 7 3=3 (mod 7) = 36 = 36 (mod Z) → 36 = 729 (mod 7) → 36 = 728 +1 (mod 7) => 36= 0+1 (mod 7) => 36 = 1 (mod 7) => (36) 76 = (1) 76 (mad 7) ⇒ 3456 = f (m07) 25 19 3 = 3 (mod 9) = 3 ² = 3² (mod af ⇒ 3² = 9 (mod af 30 (mod a) → (3²) 228 = 0 (mod: a) → 3+56=0 (mod 9): 1 11 ** 1.
a2 33 (mod 2) => 3 = 3¹ (mod) 2) 3= 2+1 (mod2) SY => 3 = 0+1. (mod 2) => 3 = 1. (mod 2) => 3456 = 1456 (mod 2) => 3456 = 1 (mod 2) Hence, the remainder of 3456 when divided by 2 is 11 b5 3=3 (mod 5) => 34 34 (mod = $₁ => 34 81 (mod 5) => 34 = 80+1 (mod 5) => 34 = 0+1 (mod 5) Đ 34 = 1 (mod 5) => (34) 114 = 1114 (mod 5) => 3456 = 1 (mod 5)
Transcribed Image Text:a2 33 (mod 2) => 3 = 3¹ (mod) 2) 3= 2+1 (mod2) SY => 3 = 0+1. (mod 2) => 3 = 1. (mod 2) => 3456 = 1456 (mod 2) => 3456 = 1 (mod 2) Hence, the remainder of 3456 when divided by 2 is 11 b5 3=3 (mod 5) => 34 34 (mod = $₁ => 34 81 (mod 5) => 34 = 80+1 (mod 5) => 34 = 0+1 (mod 5) Đ 34 = 1 (mod 5) => (34) 114 = 1114 (mod 5) => 3456 = 1 (mod 5)
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