the remainder when 3power181 is divided by 17
Answers
Answer:
3^16 = 43046721 = 1 + 17*2532160
3^181 = (3^16)^11 * 3^5
3^181 = (1 + 17*2532160)^11 * 3^5
Now, (1+an)^b = 1 + cn, for some given integers a and b, c is also an integer.
3^181 = (1 + 17c) * 3^5 = 243 + 17d = (5 + 17*14) + 17d = 5 + 17e, where d and e are also integers produced by the multiplication or addition of integers.
The remainder when 3¹⁸¹ is divided by 17 is therefore 5.
Therefore the remainder when 3¹⁸¹ is divided by 17 is 5.
Given:
The exponential power = 3¹⁸¹
To Find:
The remainder when 3¹⁸¹ is divided by 17.
Solution:
The given question can be solved as shown below.
The given exponential power = 3¹⁸¹
When 3ⁿ is divided by 17, the remainder comes from n = 3
⇒ 3³ = 27 = ( 17 + 10 )
⇒ 3¹⁸¹ = (3³)⁶⁰ × 3 = ( 17 + 10 )⁶⁰ × 3
⇒ 3¹⁸¹/17 = ( 17 + 10 )⁶⁰/17 × 3
17/17 → Remainder = 0
⇒ So 3¹⁸¹/17 = ( 10⁶⁰/17 ) × 3
( 10² )³⁰ = 100³⁰
Now 100/17 = ( 85 + 15 )/17
⇒ 100³⁰/17 × 3 = ( 85 + 15 )³⁰/17 × 3
85/17 → remainder = 0
⇒ 100³⁰/17 × 3 = ( 85 + 15 )³⁰/17 × 3 = For remainder, 15³⁰/17 × 3
(15)³⁰ = (15³)¹⁰ = (3,375)¹⁰
Now, 15³ = 3,375 = (3,366 + 9)
⇒ 15³⁰/17 × 3 = (3,366 + 9)¹⁰ × 3
3,366/17 → Remainder = 0
⇒ 15³⁰/17 × 3 = (3,366 + 9)¹⁰/17 × 3 → For remainder, 9¹⁰/17 × 3
9¹⁰ = 81⁵ = (68 + 13)⁵
⇒ 9¹⁰/17 × 3 = (68 + 13)⁵/17 × 3
68/17 → Remainder = 0
⇒ 9¹⁰/17 × 3 = (68 + 13)⁵/17 × 3 → For remainder, 13⁵/17 × 3
13⁵ = 169² × 13 = (153 + 16)² × 13
⇒ 13⁵/17 × 3 = (153 + 16)²/17 × 13 × 3
153/17 → Remainder = 0
⇒ 13⁵/17 × 3 = (153 + 16)²/17 × 13 × 3 → For remainder, (16² × 13 × 3)/17
16² = 256 = ( 255 + 1 )
⇒ (16² × 13 × 3)/17 = ( 255 + 1 )/17 × 13 × 3
255/17 → Remainder = 0
⇒ (16² × 13 × 3)/17 = ( 255 + 1 )/17 × 13 × 3 → For remainder, (1 × 13 × 3)/17
39/17 → Remainder = 5
Therefore the remainder when 3¹⁸¹ is divided by 17 is 5.
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