Frankenodd Theory Of Odd Numbers

Dr. Frankenodd (brother of Frankenine) observed the following on numbers:

  • He took 3 consecutive positive integers and he added them. He noticed that all results were divisible by 3. 3+4+5=12=3×4{\color{#3D99F6}3 + 4 + 5} = {\color{#D61F06}12} = {\color{#20A900}3 \times 4}
  • He took 5 consecutive positive integers and he added them. He noticed that all the results were divisible by 5. 43+44+45+46+47=225=5×45{\color{#D61F06}43 + 44 + 45 + 46 + 47} = {\color{#20A900}225} = {\color{#3D99F6}5 \times 45}
  • He took 7 consecutive positive integers and he added them. He noticed that all the results were divisible by 7. 22+23+24+25+26+27+28=175=7×25{\color{#20A900}22 + 23 + 24 + 25 + 26 + 27 + 28} = {\color{#3D99F6}175} = {\color{#D61F06}7 \times 25}

Final Conclusion: For nn (where nn is an odd number) consecutive positive integers the sum will always be a multiple of nn. Will Dr. Frankenodd theory always works?

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