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Prove by Induction that 3^(2n) - 5 is divisible by 4 watch

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    Hey, so I'm currently self teaching the 'Proof and elementary number theory' topic for my advanced higher but I can't seem to get my head around proof by induction. I think I understand the concept and I understand the examples in my notes, but I've only gotten a few of the questions I've tried so far. So any help with the following question would be greatly appreciated.

    Prove by induction that 3^(2n) - 5 is divisible by 7

    What I've got so far...

    First prove for n = 1,

    When n = 1, 3^(2) - 5 = 9-5 = 4, hence statement is true for n = 1

    Next, assume true for n = k where k>= 1

    => 3^(2k) - 5 = 4m for some m

    Now consider n = (k +1)

    3^(2(k+1)) - 5 = 4m <--- I think this is the goal?
    3^(2k+2) - 5 = 4m
    3^(2k) * 3^(2) - 5 = 4m

    And that's as far as I've got, what happens next?

    Cheers
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    (Original post by Big Big Ron)

    => 3^(2k) - 5 = 4m for some m

    Now consider n = (k +1)
    3^(2k) * 3^(2) - 5 = 4m
    Well, you know that 3^{2k} = 4m + 5, so:

    9\times 3^{2k} - 5 = 9(4m + 5) - 5 = \cdots can you show that \cdots is a multiple of 4?

    Edit: Luke's method is cleaner, go for that.
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    (Original post by Big Big Ron)
    Hey, so I'm currently self teaching the 'Proof and elementary number theory' topic for my advanced higher but I can't seem to get my head around proof by induction. I think I understand the concept and I understand the examples in my notes, but I've only gotten a few of the questions I've tried so far. So any help with the following question would be greatly appreciated.

    Prove by induction that 3^(2n) - 5 is divisible by 7

    What I've got so far...

    First prove for n = 1,

    When n = 1, 3^(2) - 5 = 9-5 = 4, hence statement is true for n = 1

    Next, assume true for n = k where k>= 1

    => 3^(2k) - 5 = 4m for some m

    Now consider n = (k +1)

    3^(2(k+1)) - 5 = 4m <--- I think this is the goal?
    3^(2k+2) - 5 = 4m
    3^(2k) * 3^(2) - 5 = 4m

    And that's as far as I've got, what happens next?

    Cheers
    So you've got 9*3^(2k)-5=8*3^(2k) + 1*3^(2k)-5

    Now evidently the first bit is divisible by 4, and our assumption tells us the second bit is divisible by 4, so hence the whole lot is.
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    Fantastic, I get it now. It seems totally obvious!

    Thanks a lot guys!
 
 
 
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