Maths / Proof by Contradiction

Question

Hello fellow mathematicians!

Can someone please help me solve this problem? We just started learning contradiction proofs in school and I don't quite understand it. This is the task we got as homework:

Prove by contradiction that no power of 2 is equal to the sum of any five consecutive natural numbers.

I would be very grateful if someone could answer my question. Thank you in advance! I wish you a nice rest of the day!

Boris

Answer 1

Suppose, for the sake of contradiction, there is a power of 2- call it 2ⁿ- such that 2ⁿ = m + (m + 1) + (m+ 2) + (m + 3) + (m + 4) + (m + 5). That is, 2ⁿ is the sum of five consecutive numbers. Thus 2ⁿ = 5m + 15 = 5(m + 3). So 2ⁿ is divisible by 5. However the only factors of any power of 2 are only 2, so this is a contradiction.

Thus there is no such m. As n was arbitrary, this proof holds for all natural numbers n, so there is no power of 2 that is the sum of five consecutive naturals.

Note that "5" could've been replaced by any odd integer here, and I encourage you to prove that...