Prove that if n is an odd positive integer, then x + y is a factor of xn + yn. (For example, if n = 3, then xn + yn = ( x + y )( x2 – xy + y2 ) )

Theory of Numbers (XVI) Principle of Mathematical Induction Prove that if n is an odd positive integer, then x + y is a factor of xn + yn. (For example, if n = 3, then xn + yn = ( x + y )( x2 – xy + y2 ) ) See the attached file.

Rates and Unite Prices (Basic Mathematics)

Describe a simple process for using rates and unit prices that might help someone who is having difficulty understanding these concepts. Include an example of your own to explain the solution process.

Linear Diophantine Equations

If you had an unlimited amount of 2p and 5p coins. Consider the linear diophantine equation 2x + 5y = n Show that all amounts n>=N=4 are obtainable.

Congruence

Show that if a = b (mod n) then (i) a^j = b^j (mod n) for any positive integer j (ii) ca =cb (mod n) for any integer c (iii) f(a) = f(b) (mod n) for any polynomial with integer coefficients

Continued Fractions

Show that each of the following holds: [1,1,1,1,1,1,....]^2 = [2,1,1,1,1,1,....] [1,2,2,2,2,2,....]^2 = [2] [1,1,1,1,1,1,....][0,1,1,1,1,1,....] = [1] In each case make a conjecture about a possible generalisation, and explore it (i.e. attempt to prove your conjectures true or false). Note:We can write down any continued fraction such as P/Q = a + 1/(b + 1/(c + 1/(d + ...))) just as a list of the numbers a, b, c, ... Since the first number, a, is special (it is the whole number part of the value) it is separated from the rest by a semicolon (;) and the rest are written as a list with comma separators (,) like this: P/Q = [a; b, c, d, ...]