Explanation Given information: The given expression is 89 307 ( 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 89 1 ( mod 713 ) , 89 2 ( mod 713 ) , 89 4 ( mod 713 ) , 89 8 ( mod 713 ) , 89 16 ( mod 713 ) , 89 32 ( mod 713 ) , 89 64 ( mod 713 ) , 89 128 ( mod 713 ) and 89 256 ( mod 713 ) . To find the value of 89 1 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 1 ( mod 713 ) = 89 ( mod 713 ) = 89 Thus, the value of 89 1 ( mod 713 ) is 89. To find the value of 89 2 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 2 ( mod 713 ) = 7921 ( mod 713 ) [ 7921 = 11 ⋅ 713 + 78 ] = 78 Thus, the value of 89 2 ( mod 713 ) is 78. To find the value of 89 4 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 4 ( mod 713 ) = ( 89 2 ) 2 ( mod 713 ) = [ 89 2 ( mod 713 ) ] 2 ( mod 713 ) = 78 2 ( mod 713 ) Simplify the above equation. 78 2 ( mod 713 ) = 6084 ( mod 713 ) = 380 Thus, the value of 89 4 ( mod 713 ) is 380. To find the value of 89 8 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 8 ( mod 713 ) = ( 89 4 ) 2 ( mod 713 ) = [ 89 4 ( mod 713 ) ] 2 ( mod 713 ) = ( 380 ) 2 ( mod 713 ) Simplify the above equation. ( 380 ) 2 ( mod 713 ) = 144400 ( mod 713 ) [ 144400 = 202 ⋅ 713 + 374 ] = 374 Thus, the value of 89 8 ( mod 713 ) is 374. To find the value of 89 16 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y ) . 89 16 ( mod 713 ) = ( 89 8 ) 2 ( mod 713 ) = [ 89 8 ( mod 713 ) ] 2 ( mod 713 ) = ( 374 ) 2 ( mod 713 ) Simplify the above equation. ( 374 ) 2 ( mod 713 ) = 139876 ( mod 713 ) [ 139876 = 196 ⋅ 713 + 128 ] = 128 Thus, the value of 89 16 ( mod 713 ) is 128. To find the value of 89 32 ( mod 713 ) , use the formula x 2 ( mod y ) = [ x ( mod y ) ] 2 ( mod y )

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Set Theory

Every field in the present world involves a set. The theory of sets was developed by German mathematician Georg Cantor while working on the “problems of trigonometric series”.

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Chapter 8.4, Problem 17ES

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The value of 89 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 89307(mathrmmod713).

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To calculate:

The value of 89307(mod713).

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Chapter 8 Solutions

The value of 89 307 ( mod 713 ) .

Solution Summary: The author explains how to calculate the value of 89307(mathrmmod713).

Concept explainers

To calculate: The value of 89307(mod713).

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Chapter 8 Solutions

To calculate:

The value of 89307(mod713).