Let T be an acute triangle. The area of T is denoted by A. If the length of each side of T' is an odd prime number 3 show that A2 is an integer. 16 Let T be an acute triangle. The area of T...
Problem 6: Let p be an odd prime number, so that p= 2k +1 for some positive integer k. Prove that (k!)2 = (-1)k+1 mod p. Hint: Try to see how to group the terms in the product (p − 1)! = (2k)! = 1 * 2 * 3... (2k – 2) * (2k – 1) * (2k) to get two products, each equal to k! modulo p.
Let p be an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
10.3. IUT 3 QueSLUIT HELP Let lo be an equilateral triangle with sides of length 13. The figure 14 is obtained by replacing the middle third of each 1.3 side of lo by a new outward equilateral triangle with sides of length triangle with sides of length 1 . The process is repeated where In +7 is obta ocess is repeated whe obtained 13 Answer 30 +1. Answ by replacing the middle third of each side of In by a...
) 8. Suppose a triangle is constructed where two sides have fixed length a and b, but the third side has variable length x You can imagine there is a pivot point where the sides of fixed length a and b meet, forming an angle of θ. By changing the angle θ, the opposite side will either stretch or contract (a) Let K(x)- Vs(s - a)(s -b)(s - x), where s is the semiperimeter of the triangle. Accord ing to...
3)Find the length of an altitude of an equilateral triangle if each side is 20 inches long. Question 3 Find the length of an altitude of an equilateral triangle if each side is 20 inches long. a) O 20/3 inches h b) O 105 inches c) O 20V2 inches d) O 10/3 inches e) O None of the above
3 circles tangent to each other are drawn inside a triangle. find the area inside the triangle not including the circles area. My teacher found out that it was an equilateral triangle somehow, and found out each side length of the triangle in terms of the radius of the circle. write answer in terms of radius of circle, or r. all circles are congruent
(1) The Legendre symbol and Euler's criterion. (1 pt each) Let p be an odd prime and a Z an integer which is not divisible by p. The integer a is called a quadratic residue modulo p if there is b E Z such that a b2 (p), i.e., if a has a square root modulo p. Otherwise a is called a quadratic non-residue. One defines the Legendr symbol as follows: 1 p)=T-i if a is a quadratic residue modulo...
3. Each side of an equilateral triangle measure 12 in. Find the length of an altitude of the triangle. Express the solution in simplify radicals when necessary.
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
ame: . (10 points) Let p > 3 be any prime number. (a) Show that p mod 6 is equal to 1 or 5 (b) Use part (a) to prove that pe - 1 is always a multiple of 24.