Theory
When we divide a number by m, we get a remainder between 0 and m−1. The remainder depends on the number, but for squares and cubes only certain remainders are possible. These are called the remainders (or residues) of squares and cubes modulo m.
For example, if we divide any square by 3, the remainder can only be 0 or 1. If we divide any square by 4, the remainder can only be 0 or 1 as well. This idea is useful because sometimes we can prove something is impossible just by checking all possible remainders.
Problems
- Make tables of the remainders (residues) of squares and cubes of integers modulo m for m = 3, 4, 5, 7, 8. These tables will be needed for some problems below.
- Can a number of the form 200…009 (with some number of zeros between the twos and the nines) be a perfect square of an integer for any number of zeros?
- Which digits can a perfect square end with?
- Can the square of an integer have the form
a) 5q + 2,
b) 3q − 1,
c) 6q − 1? - Take any positive integer n. Find the sum of its digits; then find the sum of digits of that result, and so on, until you get a single-digit number R.
a) Prove that R is the remainder of n upon division by 9.
b) What values can R take if n is a perfect square?
c) What values can R take if n is a perfect cube? - Does there exist a natural number N such that:
a) N² + 1 is divisible by 3?
b) N³ + 3 is divisible by 99? - Prove that if x² + y² is divisible by 3 (with x, y integers), then x and y are both divisible by 3.
- Can the sum of the squares of two odd numbers be a perfect square? What about the sum of the squares of three odd numbers?
- Let a, b, c be natural numbers with a + b + c divisible by 6. Prove that a³ + b³ + c³ is also divisible by 6.