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

1. 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
2. (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 , so:

can you show that is a multiple of 4?

Edit: Luke's method is cleaner, go for that.
3. (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.
4. Fantastic, I get it now. It seems totally obvious!

Thanks a lot guys!

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Updated: February 28, 2016
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