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

Download SGBAU (Sant Gadge Baba Amravati university) BSc 2019 Summer (Bachelor of Science) 3rd Sem Mathematics Elementary Number Theory Previous Question Paper

Time : Three Hours]
B.Sc. (PartIl) Semester III Examination
MATHEMATICS
(Elementary Number Theory)
PapcrVl
Note :(1) Question No. l is compulsory and attempt it once only.
(2) Attempt ONE question from each unit.
1. Choose the correct alternative :
(1) Two integers a and b that are not both zero are relatively prime whenever
(a) [8, b] = 1 (b) (a, b) = 1
(c) (a, b) = d, d > 1 (d) None of these 1
(2) ForneN.(n,n+l)= .
(a) 1 (b) n
(c) n + l (d) n(n + 1)
(3) A linear Diophantine equation 12x + 8y = 199 has
(a) unique solution (b) infinitely many solutions
(c) no solution (d) None of these
(4) Any two distinct Fermat numbers are .
(a) Composite (b) Relatively prime
(c) Prime numbers (d) None of these
(5) The non negative residue modulo 7 of 17 is
(a) 0 (b) 1
(c) 2 (d) 3
(6) The inverse of 2 modulo 5 is .
(a) 3 (b) 2 .
(c) 5 (d) l
(7) For any prime p, T(p) =
(a) 0 (b) 1
(c) 2 (d) None of these
(8) If n is divisible by a power of prime higher than one, then p(n) =
(a) 0 (b) 1
(c) n (d) n + 1
(9) The order of 3 modulo 5 is .
(a) 1 (b) 2
(C) 3 (d) 4
YBC15254 l
AW1683
[Maximum Marks : 60
(Contd.)

(10) A quadratic residue of 7 IS
(a) 3 (b)
4
(C) 5 (d) 6 l
UNITl
2. (a) Let % and i be fractions in [(-west terms so that (a. b) : (c, d) 1. Prove that iftheir
sum is an integer. then 1 = t d i. 4
(b) Find the gcd of 275 and 200 and express it in the form xa + yb. 4
(c) [f (a, b) == d. then show that IE, 3] = l. 2
3. (p) Prove that a common multiple of any two non zero integers a and b is a multiple of
the [cm [:1, b]. 4
(q) If (a, 4) =7 2 and (b. 4) = 2. then prove that (a 4* b. 4') = 4. 4
(r) Prove the (a, a + 2) = 1 Jr 2 for every integer a. 2
UNlTll
4 (a) If P is a prime and P | 21.: . an. then prove that P divides at least one factor a] of
the product i.e. P | aL for sum i. whcrc l S i S n. 5
(b) Find the gcd and 1cm of n 18900 and b = 17160 by writing each of the numbers a
and b in prime factorization ctmnnical form. 5
5. (p) Dene Fcrmat number. vac that thc Fcrmat number F5 is divisible by 641 and hence
is composite. 1+4
(q) Find the solution of thc lmcar Diaphantine equation 5x + By = 52. 5
UNlTlll
6 (a) Prove that congruence modulo m is an equivalence relation. 6
(b) Solve the linear congruence
15x 22 10(mod 25). 4
7. (p) Solve the system of three ct'mg'rucnccs
x E I(mod 3), x E 2(mod 5). ~: = 3(mod 7). 6
(q) If a, b, c and m are integers \xith m > 0 such that a s h(mod m). then prove that :
(ii) (a c) E (b - L) (mod m 2
(ii) ac E be (mod m). 2
UNITIV
8. (3) Dene Euler tb-function. Prove that if P is a prime and k a positive integer, then
(Pk) = P" (P 1).
Evaluate 0(3). l+3+l
YBC15254 2 (Contd.)

(b)
(C)
(q)
10. (a)
(b)
11- (p)
(Q)
If m is a positive integer and a is an integer with (a, m) = 1, then prove that
a' E 1(mod m). 3
Prove that, for any prime P,
O(P !) = (P + 1) c((P - 1)!). 2
State Mobius inversion formula.
Prove that if F is a multiplicative function and F(n) = Zd) , then f is also multiplicative.
dtn
1+4
Let n = plal p;2 ..... pr'I be the prime factorization of the integer n > 1. If f is multiplicative
function, prove that
Zu(d)f(d) =(1-ftp.))lf(p2 ..... (Hum). 5
dtn
UNlTV
If P is an odd prime number, then prove that PI has a primitive root for all positive
integer n. 5
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
Prove that the quadratic residues of odd prime P are congruent modulo P to the integers
2
13,22, ..... , P" . 5
2
Solve the quadratic congruence
5x2 6x + 2 E O(mod 13). 5
YBC15254 3 625