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Math Help - Powers of Odd Integers

  1. #1
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    Powers of Odd Integers

    Prove or disprove that the fourth power of an odd integer always leaves a remainder of 1 when divided by 16.
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    Quote Originally Posted by creativewisdom View Post
    Prove or disprove that the fourth power of an odd integer always leaves a remainder of 1 when divided by 16.
    We will prove the proposition:
    Notice (2n+1)^4 - 1 = 2n(2n+2)(4n^2+4n + 2) = 8n(n+1)(2n^2+ 2n + 1). Then observe that n(n+1) is always even. Thus 16 divides 8n(n+1). This means 16 divides (2n+1)^4 - 1.
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    Quote Originally Posted by Isomorphism View Post
    We will prove the proposition:
    Notice (2n+1)^4 - 1 = 2n(2n+2)(4n^2+4n + 2) = 8n(n+1)(2n^2+ 2n + 1). Then observe that n(n+1) is always even. Thus 16 divides 8n(n+1). This means 16 divides (2n+1)^4 - 1.

    Thank you so much for the help. My question, however, is how did you derive this proposition?
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  4. #4
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    Quote Originally Posted by creativewisdom View Post
    Thank you so much for the help. My question, however, is how did you derive this proposition?
    Well... thats what my previous post did...I derived the result. Were you not looking for a proof?
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