Section5.6Exercises

  1. Find and prove the possible last digits for a perfect square.
  2. Prove that if the sum of digits of a number is divisible by 3, then so is the number. Hint: Write 225 as \(2\cdot10^2+2\cdot 10+5\). Can you prove the same thing for 9?
  3. Give the least absolute residues and the least nonnegative residues for \(n=21\).
  4. Prove that 13 divides \(145^6+1\) and 431 divides \(2^{43}-1\) without a computer (but definitely using congruence).
  5. Compute \(7^{43}\) mod (\(11\)) as above (using powers of 2), without using Sage or anything that can actually do modular arithmetic. (You should never have to compute a number bigger than \((11-1)^2=100\), so it shouldn't be too traumatic.)
  6. Use the properties of congruence to show that if \(a\equiv b\) mod (\(n\)), then \(a^3\equiv b^3\) mod (\(n\)).
  7. Use the properties of congruence and induction to show that if \(a\equiv b\) mod (\(n\)), then \(a^m\equiv b^m\) mod (\(n\)) for any positive \(m\).
  8. For which positive integers \(m\) is \(27\equiv 5\) mod (\(m\))?
  9. Complete the proof that having the same remainder when divided by \(n\) is the same as being congruent modulo \(n\).
  10. Find some \(a\) and \(n\) such that \(a^n\) mod (6) equals \(a^{n+6}\) mod (6), where \(a\not\equiv 0,1\) and \(n\neq 0,1\). Then try to find an example where they are not equal.