SECTION 2.1 Elementary Number Theory 67
- Find all solutions of 15x+ 16y= 900, withx, y≥0.
- Suppose that someone bought a certain number of 39-cent pens
and a certain number of 69-cent pens, paying $11.37 for the total.
Find the number of 39-cent pens and the number of 69-cent pens
purchased. - I recently purchased a number of DVDs at 6Ueach and a number
of DVDs at 7Ueach, paying 249Ufor the total. Find the number of
6 UDVDs and the number of 7UDVDs assuming that I purchased
approximately the same number of each. - Solve 15x− 24 y= 3, x,y≥0.
- Farmer Jones owes Farmer Brown $10. Both are poor, and neither
has any money, but Farmer Jones has 14 cows valued at $184 each
and Farmer Jones has a large collection of pigs, each valued at
$110. Is there a way for Farmer Jones to pay off his debt? - A Pythagorean triple is a triple (a,b,c) of positive integers
such that a^2 +b^2 = c^2. Therefore, (3, 4 ,5) is an example of a
Pythagorean triple. So is (6, 8 ,10). Call a Pythagorean triple
(a,b,c) primitive ifa, b, andcshare no common factor greater
than 1. Therefore, (3, 4 ,5) is a primitive Pythagorean triple, but
(6, 8 ,10) is not.
(a) Assume thatsandtare positive integers such that
(i)t < s,
(ii)sandtare relatively prive, and
(iii) one ofs, tis odd; the other is even.
Show that ifx= 2st, y=s^2 −t^2 , z=s^2 +t^2 , then (x,y,z) is
a Pythagorean triple.
(b) Show that every Pythagorean triple occurs as in (a), above.- This problem involves a system of Diophantine equations.^6 Ed and
Sue bike at equal and constant rates. Similarly, they jog at equal
and constant rates, and they swim at equal and constant rates. Ed
(^6) Essentially Problem #3 from the 2008 American Invitational Mathematics Examination.