SOLUTIONS FOR THE TRAINING FINAL Remember : the final exam
... equations are the integers modulo 10 that are cogruent to 4̄ modulo 5, that is 4̄ and 9̄. 7.– Compute φ(18). Compute 51000 (mod 18). Solutions: φ(18) is the numebr of integer prime to 18 between 1 and 18. Those are 1, 5, 7, 11, 13, 17, so φ(18) = 6. By Euler’s theorem, since 5 is relatively prime to ...
... equations are the integers modulo 10 that are cogruent to 4̄ modulo 5, that is 4̄ and 9̄. 7.– Compute φ(18). Compute 51000 (mod 18). Solutions: φ(18) is the numebr of integer prime to 18 between 1 and 18. Those are 1, 5, 7, 11, 13, 17, so φ(18) = 6. By Euler’s theorem, since 5 is relatively prime to ...
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... Quadratic Equation: Solving by the square root method Example 2: Solve (3x – 5)2 + 36 = 0. Note: this has the form au2 + bu + c = 0, where b = 0 and u is an algebraic expression (u = 3x – 5). So first, isolate u2 = (3x – 5)2. Next, take the square root of each side and place a "" symbol in front o ...
... Quadratic Equation: Solving by the square root method Example 2: Solve (3x – 5)2 + 36 = 0. Note: this has the form au2 + bu + c = 0, where b = 0 and u is an algebraic expression (u = 3x – 5). So first, isolate u2 = (3x – 5)2. Next, take the square root of each side and place a "" symbol in front o ...
Root of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that gives 1 when raised to some positive integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, and the discrete Fourier transform.In field theory and ring theory the notion of root of unity also applies to any ring with a multiplicative identity element. Any algebraically closed field has exactly n nth roots of unity, if n is not divisible by the characteristic of the field.