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Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

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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.

89 307 mod 713

To determine

To calculate:

The value of 89307(mod713).

Explanation

Given information:

The given expression is 89307(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.

So, q=adivd and r=amodd.

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 891(mod713),892(mod713),894(mod713),898(mod713),

8916(mod713),8932(mod713),8964(mod713),89128(mod713) and 89256(mod713).

To find the value of 891(mod713), use the formula x2(mody)=[x(mody)]2(mody).

891(mod713)=89(mod713)=89

Thus, the value of 891(mod713) is 89.

To find the value of 892(mod713), use the formula x2(mody)=[x(mody)]2(mody).

892(mod713)=7921(mod713) [7921=11713+78]=78

Thus, the value of 892(mod713) is 78.

To find the value of 894(mod713), use the formula x2(mody)=[x(mody)]2(mody).

894(mod713)=( 89 2)2(mod713)=[ 892( mod713)]2(mod713)=782(mod713)

Simplify the above equation.

782(mod713)=6084(mod713)=380

Thus, the value of 894(mod713) is 380.

To find the value of 898(mod713), use the formula x2(mody)=[x(mody)]2(mody).

898(mod713)=( 89 4)2(mod713)=[ 894( mod713)]2(mod713)=(380)2(mod713)

Simplify the above equation.

(380)2(mod713)=144400(mod713) [144400=202713+374]=374

Thus, the value of 898(mod713) is 374.

To find the value of 8916(mod713), use the formula x2(mody)=[x(mody)]2(mody).

8916(mod713)=( 89 8)2(mod713)=[ 898( mod713)]2(mod713)=(374)2(mod713)

Simplify the above equation.

(374)2(mod713)=139876(mod713) [139876=196713+128]=128

Thus, the value of 8916(mod713) is 128.

To find the value of 8932(mod713), use the formula x2(mody)=[x(mody)]2(mody)

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