24.

25.

26.

27.

154396

(a)

(b)

(a)

(b)
(a)
(b)
(a)

(b)

154396

3 D 12645
Section C

Answer any two questions.
Each question carries 10 marks.

State six standard methods for proving theorems and briefly explain any two of them with the
help of examples.

Using the laws of logic simplify the Boolean Expression (p ‏ہہ‎ 9) ४4० (= DA ५).

Prove that there is no polynomial f (7) with integral coefficients that will produce primes for
all integers n.

State the prime number theorem and find six consecutive integers that are composites.
State and prove Fundamental Theorem of Arithmetic.

Find the largest power of 3 that divides 207!

Let p be a prime and a any integer such that p|a. Then show that the least residues of the
integers a, 2a,3a,. .. (9 — 1) a modulo p are a permutation of the integers

1, 2, 3,...,(p — 1).
Find the remainder when 24! is divided by 17.

(2 x 10 = 20 marks)