You are not allowed to use calculators smart phones laptops

You are not allowed to use calculators, smart phones, laptops, or any electronic devices. All you need is a paper and a pen. If you communicate with anyone during the midterm, or you use any electronic device, you will get an automatic 0 on the midterm, and it will be reported to the chair of the department. Write your student number on this booklet and on any extra sheets you hand in. Concepts Define prime numbers. Define rational numbers. State Bizout\'s Theorem. Computation Find integers s and t such that s. 21 + t. 89 = 1. Find the inverse of 89 modulo 21. Also find the inverse of 21 modulo 89. (Use your solution for the previous problem just above.) Find x = 4^123 mod 7. Show that the following statement is true: for all n ((n greaterthanorequalto 6) rightarrow (there exists x there exists y (3x + 4y = n))). Domain of the variable of the variable x, y and the non-negative integers.

Solution

1.Concepts :-

a.Prime Numbers :-

* The numbers those can be divided only by 1, or itself are known as Prime Numbers.

b.Rational Numbers :-

* The numbers those can be represented as the fraction of two integers are known as Rational Numbers.

c.Bezout\'s Theorem :-

* Let A and B be two plane curves, described by equations f(X, Y ) = 0 and g(X, Y ) = 0, where f and g are nonzero polynomials of degree m and n, respectively. Bezout’s theorem says that if all is well, then A and B meet in precisely mn points.