   Chapter 8.4, Problem 18ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 16-18, use the techniques of Example 8.4.4 and Example 8.4.5 to find the given numbers. 48 307   mod   713

To determine

To calculate:

The value of 48307(mod713).

Explanation

Given information:

The given expression is 48307(mod713).

Formula used:

Division algorithm:

Let a be an integer and d be a positive integer. Then, there are unique integers qandr with 0r<d such that a=dq+r, here, q is the quotient and r is the remainder.

Know that the technique is xm(mody)=[x(mody)]m(mody) and abmodn=[(amodn)(bmodn)](modn).

Calculation:

Write the 307 as a sum of powers of 2.

307=256+32+16+2+1

First, find the values of 481(mod713),482(mod713),484(mod713),488(mod713),

4816(mod713),4832(mod713),4864(mod713),48128(mod713) and 48256(mod713).

To find the value of 481(mod713), use the formula x2(mody)=[x(mody)]2(mody).

481(mod713)=48(mod713)=48

Thus, the value of 481(mod713) is 48.

To find the value of 482(mod713), use the formula x2(mody)=[x(mody)]2(mody).

482(mod713)=2304(mod713) [2304=3713+165]=165

Thus, the value of 482(mod713) is 165.

To find the value of 484(mod713), use the formula x2(mody)=[x(mody)]2(mody).

484(mod713)=( 48 2)2(mod713)=[ 482( mod713)]2(mod713)=1652(mod713)

Simplify the above equation.

1652(mod713)=27225(mod713) [27225=38713+131]=131

Thus, the value of 484(mod713) is 131.

To find the value of 488(mod713), use the formula x2(mody)=[x(mody)]2(mody).

488(mod713)=( 48 4)2(mod713)=[ 484( mod713)]2(mod713)=(131)2(mod713)

Simplify the above equation.

(131)2(mod713)=17161(mod713) [17161=24713+49]=49

Thus, the value of 488(mod713) is 49.

To find the value of 4816(mod713), use the formula x2(mody)=[x(mody)]2(mody).

4816(mod713)=( 48 8)2(mod713)=[ 488( mod713)]2(mod713)=(49)2(mod713)

Simplify the above equation.

(49)2(mod713)=2401(mod713) [2401=3713+262]=262

Thus, the value of 4816(mod713) is 262.

To find the value of 4832(mod713), use the formula x2(mody)=[x(mody)]2(mody)

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