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# Fermat's Little Theorem Watch

1. Hi, I have a question that I'm a bit stuck on.

PART (a)....I need to find the remainder when the number 999 999 999 992 is divided by 13.

PART (b)...Then verify this answer using Fermat's Little Theorem and the fact:
999 999 999 992 = 10^12 - 8

It mentions to verify that the hypotheses are satisfied before using the theorem.

I'm a bit stuck as I after finding the remainder to be 6 for PART (a), I am stuck on what to do for PART (b).

I have not seen any practice examples where another number is included (the "-8"). For example I have only got things like 1^6 = 1 (mod 7) to look at in my exercise book.
2. (Original post by makin)
I have not seen any practice examples where another number is included (the "-8"). For example I have only got things like 1^6 = 1 (mod 7) to look at in my exercise book.
The neat thing is that modular arithmetic is additive, so x+y (mod k) = [x (mod k) + y (mod k)] mod k.
This means you can use FLT on the individual parts. Remember that you need to verify the conditions; if you state FLT in full, you should be able to see the conditions.
3. (Original post by Smaug123)
The neat thing is that modular arithmetic is additive, so x+y (mod k) = [x (mod k) + y (mod k)] mod k.
This means you can use FLT on the individual parts. Remember that you need to verify the conditions; if you state FLT in full, you should be able to see the conditions.

Thanks for the quick reply .

So I get 10^12(mod 13) - 8(mod 13)

the 10^2 = 1(mod 13) under FLT

I'm not sure how to do the other part, if I am correct so far.
4. (Original post by makin)
Thanks for the quick reply .

So I get 10^12(mod 13) - 8(mod 13)

the 10^12 = 1(mod 13) under FLT

I'm not sure how to do the other part, if I am correct so far.
You have to state why you're allowed to say that FLT lets you reduce 10^12 to 1.
Then you've got that 10^12 - 8 = 1 - 8 (mod 13); what is (1-8) mod 13?
5. (Original post by Smaug123)
You have to state why you're allowed to say that FLT lets you reduce 10^12 to 1.
Then you've got that 10^12 - 8 = 1 - 8 (mod 13); what is (1-8) mod 13?
The rule allows me to reduce 10^12 to 1 under mod 13.

So then that goes to 1 - 8 = (mod 13) which is,

-7 = 6(mod 13) which gives us the remainder 6 which ties in with part (a).

Is that right?
6. (Original post by makin)
The rule allows me to reduce 10^12 to 1 under mod 13.

So then that goes to 1 - 8 = (mod 13) which is,

-7 = 6(mod 13) which gives us the remainder 6 which ties in with part (a).

Is that right?
Yep, that's right - when they say "verify that you're allowed to apply FLT", they mean "Write down that you've checked that 13 doesn't divide 10".
7. (Original post by Smaug123)
Yep, that's right - when they say "verify that you're allowed to apply FLT", they mean "Write down that you've checked that 13 doesn't divide 10".
Great . Really appreciate the help. It becomes so much clearer when you discuss it with someone x
8. (Original post by makin)
Great . Really appreciate the help. It becomes so much clearer when you discuss it with someone x
No problem yep, it does!

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