Answer:
(a)1≡47 mod 61
(b)1≡2329 mod 2464
(c)Does not exist
Step-by-step explanation:
The operation a(mod b) has an inverse if the the two integers (a,b)
are co-prime. i.e. their g.c.d is 1.
(a)Given 135 mod 61
We first reduce it to its lowest form.
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 rewrite 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)7465 mod 2464
Reducing it to its lowest 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)
Rewriting 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)
Therefore:
1≡-135 mod 2464
1≡(-135+2464)mod 2464
1≡2329 mod 2464
(c)42828 mod 6407
The two numbers are not co-prime. In fact their g.c.d is 43.
Therefore their inverse does not exist.
