HP 40gs hp 40gs_user's guide_English_E_HDPMSG40E07A.pdf - Page 286
Iegcdb3, Iegcd, Integer
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hp40g+.book Page 12 Friday, December 9, 2005 1:03 AM 16-12 GCD(cn,bn) = GCD(cn,2) = GCD(bn,2) = 1 Part 2 Given the equation: b3 ⋅ x + c3 ⋅ y = 1 [1] where the integers x and y are unknown and b3 and c3 are defined as in part 1 above: 1. Show that [1] has at least one solution. 2. Apply Euclid's algorithm to b3 and c3 and find a solution to [1]. 3. Find all solutions of [1]. Solution: Equation [1] must have at least one solution, as it is actually a form of Bézout's Identity. In effect, Bézout's Theorem states that if a and b are relatively prime, there exists an x and y such that: a⋅x+b⋅y = 1 Therefore, the equation b3 ⋅ x + c3 ⋅ y = 1 has at least one solution. Now enter IEGCD(B(3), C(3)). Note that the IEGCD function can be found on the INTEGER submenu of the MATH menu. Pressing a number of times returns the result shown at the right: In other words: b3 × 1000 + c3 × (-999) = 1 Therefore, we have a particular solution: x = 1000, y = -999. The rest can be done on paper: c3 = b3 + 2 , b3 = 999 × 2 + 1 Step-by-Step Examples