Download SGBAU BSc 2019 Summer 3rd Sem Mathematics Elementary Number Theory Question Paper

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B.Sc. (Part-IT) Semester-III Examination

MATHEMATICS

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(Elementary Number Theory)

Paper—VI

Time : Three Hours] [Maximum Marks : 60

Note :— (1) Question No. 1 is compulsory and attempt it once only.

(2) Attempt ONE question from each unit.

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1. Choose the correct alternative :

(1) Two integers a and b that are not both zero are relatively prime whenever

(a) [ab]=1 (b) (a,b)=1

(c) (a,b)=d,d>1 (d) None of these 1

(2) For n∈N, (n,n+1)=

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(a) 1 (b) n

(c) n+1 (d) n(n+1) 1

(3) A linear Diophantine equation 12x + 8y = 199 has

(a) unique solution (b) infinitely many solutions

(c) no solution (d) None of these 1

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(4) Any two distinct Fermat numbers are

(a) Composite (b) Relatively prime

(c) Prime numbers (d) None of these 1

(5) The non negative residue modulo 7 of 17 is

(a) 0 (b) 1

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(c) 2 (d) 3 1

(6) The inverse of 2 modulo 5 is .

(a) 3 (b) 2

(c) 5 (d) 1 1

(7) For any prime p, τ(p) =

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(a) 0 (b) 1

(c) 2 (d) None of these 1

(8) If n is divisible by a power of prime higher than one, then φ(n) =

(a) 0 (b) 1

(c) n (d) n+1 1

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(9) The order of 3 modulo 5 is

(a) 1 (b) 2

(c) 3 (d) 4 1

(10) If 28 = 1(mod 9), then the order of 2 modulo 9 is

(a) 3 (b) 4

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(c) 5 (d) 6 1

UNIT—I

2. (a) Let a/b and c/d be fractions in lowest terms so that (a, b) = (c, d) = 1. Prove that if their sum is an integer, then | b | =|d |. 4

(b) Find the gcd of 275 and -200 and express it in the form xa + yb. 4

(c) If (a, b) = d, then show that (a/d, b/d) = 1. 2

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3. (p) Prove that a common multiple of any two non zero integers a and b is a multiple of the lcm [a, b). 4

(q) If (a, 4) = 2 and (b, 4) = 2, then prove that (a + b, 4) = 4. 4

(r) Prove the (a, a + 2) = 1 or 2 for every integer a. 2

UNIT—II

4. (a) If P is a prime and P|a₁a₂ ... a_n, then prove that P divides at least one factor a_i of the product i.e. P | a_i for some i, where 1

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(b) Find the gcd and lcm of a = 18900 and b = 17160 by writing each of the numbers a and b in prime factorization canonical form. 5

5. (p) Define Fermat number. Prove that the Fermat number F₅ is divisible by 641 and hence is composite. 1+4

(q) Find the solution of the linear Diaphantine equation 5x + 3y = 52. 5

UNIT—III

6. (a) Prove that congruence modulo m is an equivalence relation. 6

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(b) Solve the linear congruence

15x ≡ 10(mod 25). 4

7. (p) Solve the system of three congruences

X ≡ 1(mod 3), x ≡ 2(mod 5), x ≡ 3(mod 7). 6

(q) If a, b, c and m are integers with m > 0 such that a ≡ b(mod m), then prove that :

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(i) (a+c)≡(b+c)(mod m) 2

(ii) ac ≡ bc (mod m). 2

UNIT—IV

8. (a) Define Euler φ-function. Prove that if P is a prime and k a positive integer, then φ(P^k) = P^k - P^k-1.

Evaluate φ(3). 1+3+1

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9. (a) If m is a positive integer and a is an integer with (a, m) = 1, then prove that

a^φ(m) ≡ 1(mod m). 3

(b) Prove that, for any prime P,

σ(P^k)= (P^k+1-1)/(P-1). 2

10. (a) State Mobius inversion formula.

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(b) Prove that if F is a multiplicative function and F(n) = Σf(d) , then f is also multiplicative. 1+4

d|n

11. (p) Let n=p₁^a1 p₂^a2..... p_r^ar be the prime factorization of the integer n > 1. If f is multiplicative function, prove that

Σf(d) = (1+ f(p₁))(1+ f(p₂))...(1+ f(p_r)) 5

d|n

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UNIT—V

12. (a) If P is an odd prime number, then prove that Pⁿ has a primitive root for all positive integer n. 5

(b) Define the order of a modulo m. Given that a has order 3 modulo P, where P is an odd prime, show that a + 1 must have order 6 modulo P. 1+4

13. (p) Prove that the quadratic residues of odd prime P are congruent modulo P to the integers

1²,2², ... ((P-1)/2)² 5

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(q) Solve the quadratic congruence

5x² — 6x + 2 ≡ 0(mod 13). 5

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