Deduce that ## 13\mid (11^{12n+6}+1) ##

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In summary, applying Fermat's theorem shows that for any integer n ≥ 0, if a = 11, p = 13, and p does not divide a, then 13 divides (11^(12n+6) + 1). This is also true for any other integer n ≥ 0 and can be proven by induction, although using Fermat's theorem is easier.
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Homework Statement
From Fermat's theorem deduce that, for any integer ## n\geq 0, 13\mid (11^{12n+6}+1) ##.
Relevant Equations
None.
Proof:

Let ## n\geq 0 ## be any integer.
Applying the Fermat's theorem produces:
## a=11, p=13 ## and ## p\nmid a ##.
Then ## 11^{13-1}\equiv 1\pmod {13}\implies 11^{12}\equiv 1\pmod {13} ##.
Observe that
\begin{align*}
&11^{12n+6}+1\equiv [(11^{12})^{n}\cdot 11^{6}+1]\pmod {13}\\
&\equiv [1^{n}(-2)^{6}+1]\pmod {13}\\
&\equiv (64+1)\pmod {13}\\
&\equiv 65\pmod {13}\\
&\equiv 0\pmod {13}.\\
\end{align*}
Thus ## 13\mid (11^{12n+6}+1) ##.
Therefore, ## 13\mid (11^{12n+6}+1) ## for any integer ## n\geq 0 ##.
 
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Correct. If you want to try, you could probably try a proof by induction, too.
 
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fresh_42 said:
Correct. If you want to try, you could probably try a proof by induction, too.
True, but I like using Fermat's theorem more. It's easier than proof by induction. I do want to admit that, I definitely need to practice more about writing proofs by induction, since I am not good at it.
 

FAQ: Deduce that ## 13\mid (11^{12n+6}+1) ##

1. How do you deduce that 13 divides (11^(12n+6)+1)?

To deduce that 13 divides (11^(12n+6)+1), we can use the divisibility rule for 13. According to this rule, for any integer a, if the alternating sum of its digits is divisible by 13, then a is also divisible by 13. In this case, we can rewrite (11^(12n+6)+1) as (11^(12n+6)-(-1)), which has an alternating sum of digits equal to 12n+7. Since 12n+7 is divisible by 13, we can conclude that 13 divides (11^(12n+6)+1).

2. What is the significance of 11^(12n+6) in the expression (11^(12n+6)+1)?

The term 11^(12n+6) is significant because it is the base of the expression and it is raised to a power that is a multiple of 12. This means that when we expand the expression using the binomial theorem, all the terms except the last one will be divisible by 13. This makes it easier to see that 13 divides the entire expression.

3. Can you provide an example to illustrate the divisibility of (11^(12n+6)+1) by 13?

Sure, let's take n=1 as an example. In this case, (11^(12n+6)+1) becomes (11^(18)+1), which is equal to 285311670611. The alternating sum of digits in this number is 5-8+3-1+0-6+1-1+6-7+0-1 = -7, which is divisible by 13. Therefore, 13 divides (11^(18)+1) and by extension, (11^(12n+6)+1) for any value of n.

4. How does the divisibility of (11^(12n+6)+1) by 13 relate to patterns in the powers of 11?

The divisibility of (11^(12n+6)+1) by 13 relates to patterns in the powers of 11 because every 12th power of 11 will have an alternating sum of digits equal to -1, which is equivalent to 12 (mod 13). This means that every 12th power of 11 will be congruent to -1 (mod 13). Since (12n+6) is a multiple of 12, (11^(12n+6)+1) will also be congruent to -1 (mod 13), making it divisible by 13.

5. What other numbers besides 13 have similar divisibility patterns with powers of 11?

There are other numbers besides 13 that have similar divisibility patterns with powers of 11. For example, 3 and 37 also have similar patterns. The divisibility rule for 3 states that if the sum of the digits of a number is divisible by 3, then the number itself is also divisible by 3. Since the sum of digits in (11^(12n+6)+1) is always 12, which is divisible by 3, we can conclude that 3 also divides (11^(12n+6)+1). Similarly, the divisibility rule for 37 states that if the alternating sum of digits of a number is divisible by 37, then the number itself is also divisible by 37. Since the alternating sum of digits in (11^(12n+6)+1) is always 12n+7, which is divisible by 37, we can conclude that 37 also divides (11^(12n+6)+1).

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