HP 48gII hp 48gII_user's manual_English_E_HDPMSG48E67_V2.pdf - Page 190

The CHINREM function CHINREM stands for CHINese REMainder. The operation coded in this command solves a system of two congruences using the Chinese Remainder Theorem. This command can be used with polynomials, as well as with integer numbers (function ICHINREM). The input consists of two vectors [expression_1, modulo_1] and [expression_2, modulo_2]. The output is a vector containing [expression_3, modulo_3], where modulo_3 is related to the product (modulo_1)⋅(modulo_2). Example: CHINREM(['X+1', 'X^2- 1'],['X+1','X^2']) = ['X+1',-(X^4-X^2)] Statement of the Chinese Remainder Theorem for integers If m1, m2,...,mr are natural numbers every pair of which are relatively prime, and a1, a2, ..., ar are any integers, then there is an integer x that simultaneously satisfies the congruences: x ≡ a1 (mod m1), x ≡ a2 (mod m2), ..., x ≡ ar (mod mr). Additionally, if x = a is any solution then all other solutions are congruent to a modulo equal to the product m1⋅m2⋅ ... mr. The EGCD function EGCD stands for Extended Greatest Common Divisor. Given two polynomials, A(X) and B(X), function EGCD produces the polynomials C(X), U(X), and V(X), so that C(X) = U(X)*A(X) + V(X)*B(X). For example, for A(X) = X^2+1, B(X) = X^2-1, EGCD(A(X),B(X)) = {2, 1, -1}. i.e., 2 = 1*( X^2+1')-1*( X^2-1). Also, EGCD('X^3-2*X+5','X') = { 5, '-(X^2-2)', 1}, i.e., 5 = - (X^2-2)*X + 1*(X^32*X+5). The GCD function The function GCD (Greatest Common Denominator) can be used to obtain the greatest common denominator of two polynomials or of two lists of polynomials of the same length. The two polynomials or lists of polynomials will be placed in stack levels 2 and 1 before using GCD. The results will be a polynomial or a list representing the greatest common denominator of the two polynomials or of each list of polynomials. Examples, in RPN mode, follow (calculator set to Exact mode): 'X^3-1'`'X^2-1'`GCD Results in: 'X-1' {'X^2+2*X+1','X^3+X^2'} ` {'X^3+1','X^2+1'} ` GCD results in {'X+1' 1} Page 5-19