Govt. M. M. college, Jessore
Dept. of Mathematics
M.Sc 1st Test Examination-2008
Sub-Theory of Numbers Code-3705
Full Marks-50 Time-2.5 hours
[All questions are of equal value. Answer any four questions ]
- (a) State and prove unique of Factorization theorem of Arithmetic.
(b) Define number of divisors and sum of divisors. Find the number of divisors
and sum of divisors of 14553 .
(c) If p is a prime number and p|ab then show that p|a or p|b .
- (a) Define multiplicative function. Show that Euler f – function is multiplicative.
(b) Find the integer solution of the Diophantine equation 56x+72y =40.
- (a) State and prove Wilson’s theorem . Write it’s converse.
(b) Using congruence show that the number is divisible by 97.
- (a) If f is a multiplicative function then its Dirichlet’s inverse function is also a
(b) Define Ù(n) . Show that (i) (ii) .
(a) If f is multiplicative function then prove that g(n)=
is also multiplicative function. (b) Prove that =
- (a) Define quadratic field and quadratic integer. Find all quadratic integers of the
quadratic field of .
(b) If both m and n are expressible as the sum of two squares then prove that mn is
also expressible as the sum of two squares.