Number Theory Homework (2 Weeks)

Homework problems for 5 Nov 2011:

Let a₁, a₂, … be a series recursively defined as follows: a₁ = 20, a₂ = 11, and for n ≥ 1, a_n+2 is the remainder when a_n+1 + a_n is divided by 100. What is the remainder when $a_1^2 + a_2^2 + \dots + a_{2011}^2$ is divided by 8?
(IMO 1962 Q1) Find the smallest positive integer with the following properties:

in decimal representation, its last digit is 6;
if we move this digit to the front of the number, we get 4 times the original number.

Find two 3-digit numbers (abc) and (def) such that upon placing them side-by-side, we get a 6-digit number (abcdef) satisfying : (abcdef) = ( (abc) + (def) )².
(a) A positive integer is written on the board. At each turn, we erase its unit digit and add 4 times that digit to the remaining number. Starting from 5²⁰¹¹, can we get 2011⁵ at some point? (b) Suppose instead, at each turn, we erase its unit digit and add 5 times that digit to the remaining number. Starting from 7²⁰¹¹, can we get 2011⁷ at some point?
Let a₁, a₂, … be a series recursively defined: a₀ = 1, a₁ = 3, and for n ≥ 0, a_n+2 = 6a_n+1 – a_n. Prove that for each n, either $\frac{a_n+1} 2$ or $\frac{a_n-1}2$ is a perfect square.

Homework problems for 12 Nov 2011:

Prove that if a, n are integers and n > 0, then whenever $a \equiv 1 \pmod n$ , we have $a^n \equiv 1 \pmod {n^2}$ .
Classify the solutions of $\frac 1 x + \frac 1 y = \frac 1 z$ , where x, y and z are positive integers.
Prove that if n =3^k, then n | (2ⁿ + 1). In particular, there are infinitely many n such that n | (2ⁿ + 1). Can you find an n with at least 2 distinct prime factors such that n | (2ⁿ + 1)?
(IMO 1971 shortlisted) Find all integer solutions to $x^2 + y^2 = (x-y)^3$ .
Suppose n is a positive integer such that n | (2ⁿ + 1). Prove that n = 1 or n is a multiple of 3.

[ Coming attractions : quadratic residues… stay tuned! 🙂 ]