Theory of Numbers Code-3705 M.Sc 1st Test Examination-2008

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 ]


  1. (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 .

  1. (a) Define multiplicative function. Show that Euler f – function is multiplicative.

(b) Find the integer solution of the Diophantine equation 56x+72y =40.

  1. (a) State and prove Wilson’s theorem . Write it’s converse.

(b) Using congruence show that the number is divisible by 97.

  1. (a) If f is a multiplicative function then its Dirichlet’s inverse function is also a

multiplicative function.

(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   =

  1. (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.


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