Show that there is a positive integer no such that for any i

Show that there is a positive integer no such that for any integer n > there nonnegative integers x and y such that n = 6x + 7y. Show that for any integer n, there exist integers x and y such that n = 6x + 7y. Use the modular exponentiation algorithm to find 2^77 mod 3. Find the inverse of 23 modulo 24. Show that Exists xP(x) wedge Exists xQ(x) and Exists x(P(x) wedge Q(x)) are not logically equivalent.

Solution

3.

Given n = 6x + 7y, thus   n = 6(x+y) + y = 6z + y

Now z = n/6 and y = n%6, and that does the job ;) find x from z and y

4.

2⁷⁷mod3 =   (2³⁸mod3)(2³⁹mod3) mod 3   = (2³⁸mod3)² (2mod3) mod 3

     = (2¹⁹mod3)⁴(2 mod 3) mod 3  

     = [ (2¹⁰mod3 . 2⁹mod3 )⁴(2 mod 3) ] mod 3

     =   [ (1.2)⁴(2) ] mod 3 = 32 mod 3 = 2

5.

Inverse of 23 modulo 24

Using Euler\'s theorem,   gcd( 23, 24 ) = 1 as 23 is a prime number

Now, Euler totient value( 24 ) = 24*( 1 - 1/2 )*(1 - 1/3 ) = 8

So, we have that 23^-1 mod 24 = 23^8-1mod 24 = 23⁷mod 24 =23

6.

Say x belongs to X ( some set, whose elements we give as input on P and Q )

First equation is true only when any one element from X satisfies P and some other element satisfies Q. That is, it is not necessary that some element has to satisfy both P and Q.

Second is true only when some element from X, satisfies both P and Q

Clearly not logically equivalent