**[OBJECTIVE]**

**Subject:** Elementary Number Theory

**Time Allowed: **10 Minutes

**Maximum Marks**: 10

**NOTE:** Attempt this Paper on this Question Sheet only. Please encircle the correct option. Division of marks is given in front of each question. This Paper will be collected back after expiry of time limit mentioned above.

__Part-I Encircle the correct option, cutting and overwriting is not allowed. (10)__

- The remainder obtained after dividing 375° by 100 is

(a). 1 (b). 11

(c). 21 (d). none of them

- If gcd(a, b) = 1 then gcd(a
^{n}, b^{n}) = _____

(a). a^{n} (b). a

(c). 1 (d). ab

- How many integers are co-prime to ‘50’

(a). 40 (b). 49

(c). infinitely many (d). none of them

- The integer that does not have order modulo 8 is

(a). 3 (b). 4

(c). 5 (d). none of them

- If p, is the nth prime number, then pn ______ 2
^{2n-1}

{a). < (b). >

(c). = (d). none of them

- Which of the following is a quadratic non-residue of 23

(a). 2 (b). 3

(c). 4 (d). none of them

- The number of zeroes present in 15! are

(a). 1 (b). 2

(c). 3 (d). none of them

- A quadratic congruence always has

(a). no solution (b). 1 solution

(c). 2 solutions (d). none of them

- If 2 is prime then

(a). an-1 = 1 (mod n) (b). a (^{n}) = 1 (mod n)

(c). (n – 1)! = —1 (mod n) (d). none of them

- For any integer n, the smallest integer that divides n(n + 1)(2n + 1) is

(a). 3 (b). 6

(c). 9 (d). none of them

** **

**[SUBJECTIVE]**

**Subject:** Elementary Number Theory

**Time Allowed: **2 Hours 45 Minutes

**Maximum Marks**: 50

**NOTE:** ATTEMPT THIS (SUBJECTIVE) ON THE SEPARATE ANSWER SHEET PROVIDED.

__Part-II Give short details of each of them, each answer carries equal marks. (20)__

**Q#**1: Use Euclidean Algorithm to find a solution of the equation 56X + 72Y = 8.

**Q#**2: Is there exists an integer @ such that 15 divides 6a – 1? Justify your answer.

**Q#**3: If a = b(modm) then show that a^{n} = b^{n} (modm) for all n ≥ 1.

**Q#**4: Describe (without proof) what you know about the solution of the equation aX + bY = c.

__ __

__Part-III Give brief answers, each answer carries equal marks. (30)__

**Q#**1: Show that the linear congruence aX = b (mod n) has a solution if and only if d|b, where d = (a, n).

**Q#**2: Find the remainders when 2^{50} and 41^{65} are divided by 7.

**Q#**3: If a cock is worth 5 coins, a hen 3 coins, and three chicks together 2 coin, how many cocks, hens, and chicks, totaling 100, can be bought for 100 coins?

**Q#**4: Let a = a_{n-1}a^{10-1} + a_{n-2}a^{10-2} +… +a_{1}10 + a_{0} be the decimal representation, so that we write a as a sequence a_{r}a_{r}-1..a_{1}a_{0} then show that 13|a if and only if 13|a_{r}a_{r}-1 –a_{1} — 9a_{0}

**Q#**5: Solve the following system: x ≡ 5 (mod 6), x ≡ 4 (mod 11), x ≡ 3 {mod 17).