5 pages Test 2, Math 3510 11/13/15 Solutions Name (9) 1. Find the prime factorization of the integer 4290. The digits sum to 15 and so 4290 has 30 as a factor and, of course 90 = (2)32(5). Therefore 12870 = (2)32(5)(143) = 213251111131 2. Determine whether the following constructions are possible with a straight edge and compass alone and in finitely many steps and assuming a unit length is given. (15) (Mark each choice yes if it is possible and no otherwise.) (a) A regular polygon with 10 sides (a decagon) inscribed in a circle of radius one (unit). Yes, because we can construct a pentagon and bisect the angles between vertices. (b) A regular polygon with 7 sides inscribed in a circle of radius one (unit). No. (c) A line segment of length 21/3 units. No. This was a assigned problem and it corresponds to “doubling the cube” (d) A line segment of length 3+21/4 units. Yes, we can construct 3 and 2 and we can take the square root and so we can take the fourth root. (e) Given an angle of arbitrary size, say θ degrees, construct another angle of θ/3 degrees. No. This is the trisection of the angle. In particular the 60 degree angle cannot be trisected. 3. Suppose that n 2 is a integer with the property that, n has no prime factors p (11) such that p √≥n. Explain why n must be prime. ≤ We know that n has a prime factorization. Suppose that prime factorization has two primes in it p and q. Both p> √n and q > √n by the given property and so n pq > √n√n = n and this is impossible and so n has only one prime factor and≥ it must be n itself: n is prime. (13) 4. (a) Use the Euclidean algorithm to find the gcd(2622,1104). Find gcd(2622,1104) by the Euclidean algorithm 2622 = 2(1104)+414 1104 = 2(414)+276 414 = 276+138 276 = 2(138)

so that 138 = gcd(2622,1104). 2

(9) (b) Obtain integers x and y so that gcd(2622,1104)= 2622x + 1104y From Part (a), gcd(2622,1104)=138 = 414-276=414-(1104-2(414))=3(414)- 1104 =3(2622-2(1104))-1104= 3(2622)-7(1104). 5. State or describe the Euclid’s fifth postulate. Any equivalent version of the postu- (7) late (such as Playfair’s) is acceptable. The original fifth postulate stated: If a straight line falling on two straight lines makes interior angles on the same side less than two right angles, then the two straight lines, if produced indefinitely meet on that side which the angles are less than two right angles. Playfair’s equivalent postulate was: Through a given point not on a given line only one parallel can be drawn to the given line. (14) 6. Discuss Euclid’s Elements. Some ideas that might be included(this list does not constitute an essay) Written (by Euclid) about 300 BCE. 13 books • First book, to our knowledge to derive much of planar geometry from a start- • ing point of postulates. 5 postulates. • the fifth postulate was controversial. It was later proved to be independent of • the other postulates and it was not valid when considering geometry on the sphere with straight lines replaced by great circle routes. some of the derivations used pictures that did not reflect the postulates and • lead to results that could not be logically derived from the postulates. Pictures proofs were discredited. Number theory appeared in the later books (VII, VIII and IX) • the prime factorization theorem and that there are infinitely many primes. • the Elements went to over 1000 editions and was used as a textbook for over • 2000 years. Immanuel Kant cited its geometry as an example of absolute truth. • (14) 7. Discuss Archimedes, emphasizing his mathematics. Again the following list does not constitute an essay but suggests some topics worth including. He solved the King Hieron’s puzzle of the crown of debased gold. In the • bathtub Archimedes realized that the water his body displaced had the same volume as his body. Eureka! He wrote Measurement of the Circle where he estimated π and also proved • the formula for the area of the circle. He showed that a sphere circumscribed by a right circular cylinder had 2/3 • the volume of the cylinder. The picture was engraved on hiss tombstone. 3

He anticipated calculus by his method of finding the area cut from a parabola • by a chord. (The volume of the sphere in terms of the cylinder above also anticipates calculus) He constructed war machines to defend the people of Syracuse from the at- • tacking Romans. He contructed catapults and huge hooks to lift ships from the water.

8. Discuss an ancient Greek mathematician (other than Archimedes, Euclid Pythago- (9) ras or Thales.) Eratosthenes invented the mesolabium to “double the cube” He invented the sieve method for finding all primes below a given number, efficiently. Eudoxus of Cnidos estimated π by finding the circumference of polygons that approximated teh cir- cle. Appolonius wrote his text, Conic Sections. Hypatia wrote commentary or edited Diophantus’s Arithmeticae and Appollonius’s Conic Sections and Claudius Ptolemy’s Almagest. She lectured at the Museum in Alexandria. She fell afoul of the early Christians who thought that she espoused paganism in her lectures and they waylaid and killed her.