Explanation Given information: The given expression is 48 307 ( mod 713 ) . Formula used: Division algorithm: Let a be an integer and d be a positive integer. Then, there are unique integers q and r with 0 ≤ r < d such that a = d q + r , here, q is the quotient and r is the remainder. So, q = a div d and r = a mod d . Know that the technique is x m ( mod y ) = [ x ( mod y ) ] m ( mod y ) and a b mod n = [ ( a mod n ) ( b mod n ) ] ( mod n ) . Calculation: Write the 307 as a sum of powers of 2. 307 = 256 + 32 + 16 + 2 + 1 First, find the values of 48 1 ( mod 713 ) , 48 2 ( mod 713 ) , 48 4 ( mod 713 ) , 48 8 ( mod 713 ) , 48 16 ( mod 713 ) , 48 32 ( mod 713 ) , 48 64 ( mod 713 ) , 48 128 ( mod 713 ) and 48 256 ( mod 713 ) . To find the value of 48 1 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 48 1 ( mod 713 ) = 48 ( mod 713 ) = 48 Thus, the value of 48 1 ( mod 713 ) is 48. To find the value of 48 2 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 48 2 ( mod 713 ) = 2304 ( mod 713 ) [ 2304 = 3 ⋅ 713 + 165 ] = 165 Thus, the value of 48 2 ( mod 713 ) is 165. To find the value of 48 4 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 48 4 ( mod 713 ) = ( 48 2 ) 2 ( mod 713 ) = [ 48 2 ( mod 713 ) ] 2 ( mod 713 ) = 165 2 ( mod 713 ) Simplify the above equation. 165 2 ( mod 713 ) = 27225 ( mod 713 ) [ 27225 = 38 ⋅ 713 + 131 ] = 131 Thus, the value of 48 4 ( mod 713 ) is 131. To find the value of 48 8 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 48 8 ( mod 713 ) = ( 48 4 ) 2 ( mod 713 ) = [ 48 4 ( mod 713 ) ] 2 ( mod 713 ) = ( 131 ) 2 ( mod 713 ) Simplify the above equation. ( 131 ) 2 ( mod 713 ) = 17161 ( mod 713 ) [ 17161 = 24 ⋅ 713 + 49 ] = 49 Thus, the value of 48 8 ( mod 713 ) is 49. To find the value of 48 16 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 48 16 ( mod 713 ) = ( 48 8 ) 2 ( mod 713 ) = [ 48 8 ( mod 713 ) ] 2 ( mod 713 ) = ( 49 ) 2 ( mod 713 ) Simplify the above equation. ( 49 ) 2 ( mod 713 ) = 2401 ( mod 713 ) [ 2401 = 3 ⋅ 713 + 262 ] = 262 Thus, the value of 48 16 ( mod 713 ) is 262. To find the value of 48 32 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y )

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Author: EPP

Publisher: CENGAGE L

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Chapter 8.4, Problem 18ES

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The value of 48 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 48307(mathrmmod713).

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To calculate:

The value of 48307(mod713).

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Chapter 8 Solutions

The value of 48 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 48307(mathrmmod713).

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To calculate: The value of 48307(mod713).

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Chapter 8 Solutions

To calculate:

The value of 48307(mod713).