Question 146
Numbers that are positive integers squared are called perfect squares; for example, 16 is a perfect square because it equals $4^{2}$. A curious fact is that perfect squares leave only remainders 0 or 1 when divided by 4. With this we can prove, for example, that 2014 is not a perfect square because 2014 leaves remainder 2 when divided by 4.
a) Knowing that every odd integer has the form $2k+1$, show that odd perfect squares leave remainder 1 when divided by 8.
b) Is it possible to place 45 odd integers in a sequence so that the sum of any 5 consecutive ones and the sum of any 9 consecutive ones are perfect squares?
a) Knowing that every odd integer has the form $2k+1$, show that odd perfect squares leave remainder 1 when divided by 8.
b) Is it possible to place 45 odd integers in a sequence so that the sum of any 5 consecutive ones and the sum of any 9 consecutive ones are perfect squares?