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1 month ago

Using the extended Euclidean algorithm, find the multiplicative inverse of a. 135 mod 61 b. 7465 mod 2464 c. 42828 mod 6407

Mathematics

2 answers:

Zina [12.3K]1 month ago

6 0

Answer:

a. 47

b. 2329

c. No co-prime relationship exists.

Step-by-step explanation:

a. 135 mod 61

The solution is provided.

b. 7465 mod 2464

2329

The solution is provided.

c. 42828 mod 6407

The numbers n and p are not co-prime.

Zina [12.3K]1 month ago

4 0

Answer:

(a) 1≡47 mod 61

(b) 1≡2329 mod 2464

Step-by-step explanation:

The expression a(mod b) possesses an inverse if the two integers (a,b) are co-prime, meaning their greatest common divisor (g.c.d) is 1.

(a) For 135 mod 61

We need to simplify it initially.

135 mod 61=13 mod 61

61=13(4)+9 ==> 9=61-13(4)

13=9(1)+4 ==> 4=13-9(1)

9=4(2)+1 ==> 1=9-4(2)

4=1(4)

Next, we will express 1 as a linear combination of 13 and 61.

1=9-4(2)

=9-(13-9(1))2

=9(3)-13(2)

=(61-13(4))(3)-13(2)

=61(3)-13(12)-13(2)

1=61(3)-13(14)

1=61(3)+13(-14)

1≡-14 mod 61≡(-14+61)mod 61

1≡47 mod 61

(b) For 7465 mod 2464

Begin by reducing it to the simplest form

7465 mod 2464=73 mod 2464

2464=73(33)+55 ==>55=2464-73(33)

73=55(1)+18 ==> 18=73-55(1)

55=18(3)+1 ==>1=55-18(3)

18=1(18)

Now expressing 1 as a linear combination of 73 and 2464.

1=55-18(3)

=2464-73(33)-(73-55(1))(3)

=2464-73(33)-73(3)+55(3)

=2464-73(36)+55(3)

=2464-73(36)+(2464-73(33))(3)

=2464-73(36)+2464(3)-73(99)

=2464(4)-73(135)

1=2464(4)+73(-135)

Thus:

1≡-135 mod 2464

1≡(-135+2464)mod 2464

1≡2329 mod 2464

The two integers are not co-prime, having a g.c.d of 43.

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1 month ago

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