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2 D 32372

Section B

Answer any number of questions.
Each question carries 5 marks.
Maximum 35 marks.

Prove that an implication is logically equivalent to its contra positive.
Let b be an integer > 2. Suppose ¢ + 1 integers are randomly selected. Prove that the
difference of two of them is divisible by த்‌.

For every positive integer n, prove that there are n consecutive integers that are composite
numbers.

Let f,, denote the nth Fermat number. Prove that 641|/,.

Let a and b be any positive integers, and r the remainder, when a is divided by b. Prove
that (a, b) = (9, 7).

ab

Let a and b be positive integers. Then [a,b] = 7.

Find the remainder when 16° is divided by 7.

Let m be a positive integer and a any integer with (a, m) = 1. Prove that a(m)=1(modm).
(Maximum ceiling 35 marks)
Section C

Answer any two questions.
Each question carries 10 marks.

(a) Using the laws of logic, prove that [(qv p) (qv ~@|v ~ pPAM=Pv 4.

(b) By using the method of contradiction, prove that there is a prime number > 3.
(a) Prove that the function is multiplicative.

(b) Evaluate (1976).

و
‎p-l‏

(a) Find the prime p for which is a square.

(b) Solve the linear congruence 12x = 6(mod7).

(a) Prove that there are infinitely many primes of the form 4n + 3.

(9) Prove that the product of any two integers of the form 4n + 1 is also of the same form.
(2 x 10 = 20 marks)

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