A number b divides a if the remainder is zero. We denote it by b | a. b a denotes that b does not divide a. If b divides a then a is a multiple of b. Now we can define the greatest common divisor (GCD). The GCD of two numbers a and b is defined as the biggest number which divides both a as well as b. It is also denoted by gcd(a, b). One of the important case is when gcd(a, b) = 1, i.e., there i...

We study the approximability of the following NP-complete (in their feasibility recognition forms) number theoretic optimization problems: 1. x 2 Z n with minimum max 1in jxij satisfying P n i=1 xiai = gcd(a1; : : : ; an). We present a polynomial-time algorithm which approximates a minimum gcd set for a1; : : : ; an within a factor 1+ln n and prove that this algorithm is best possible in the se...

This note is a response to one of the problems posed by Kwa´sniewski in [1, 2], see also [3] i.e. GCD-morphic Problem III. We show that any GCD-morphic sequence F is at the point product of primary GCD-morphic sequences and any GCD-morphic sequence is encoded by natural number valued sequence satisfying condition (C1). The problem of general importance-for example in number theory was formulate...

Received Jul 17 th , 2012 Accepted Aug 26 th , 2012 Recently, many fully-homomorphic encryption schemes have been constructed. However, the issue of the security of these fully homomorphic encryptions has not been carefully studied. By using lattice reduction algorithm, we firstly present an attack on the fully homomorphic encryption based on approximate GCD over the integers. Our result shows ...

Euclid’s algorithm for computing the greatest common divisor of 2 numbers is considered to be the oldest proper algorithm known ([10]). This algorithm can be amplified naturally in various ways. The GCD problem for more than two numbers is interesting in its own right. Thus, we can use Euclid’s algorithm recursively to compute the GCD of more than two numbers. Also, we can do a constructive com...

PURPOSE To investigate longitudinal changes in goblet cell density (GCD) in contact lens (CL) wearers who do and do not develop symptoms of dry eye (DE). METHODS Sixty healthy individuals fitted with daily disposable hydrogel CLs and 23 age-balanced non-CL-wearing controls underwent assessment using the 5-item dry eye questionnaire, noninvasive tear film break-up time measurement, ocular surf...