Determine all positive integers n for which 13^n + 6 is divisible by 7. Prove th

ID: 3210354 • Letter: D

Question

Determine all positive integers n for which 13^n + 6 is divisible by 7. Prove that your answer is correct

The procedure of proof should follow the following format. The answer should like that.

Example

Prove:For all integers a,b,c,m where m > 0, if ab (mod m) and bc (mod m),then ac (mod m)

Proof. Let a,b,c,m be arbitrary integers, suppose that m > 0.

(1)Suppose that ab (mod m) and bc (mod m)

(2)From (1) and using a result from last class, we conclude that m(a-b) and m(b-c)

(3)From (2) and the definition of divides, a-b = mk1and b-c = mk2for some integers k1and k2

(4)From (3), we write (a-b) + (b-c) = mk1+ mk2= m(k1+ k2),which means that a - c = m(k1+ k2). By definition of divides, m(a-c)

(5)From (4) and a result from last class, m(a-c) implies that ac (mod m).

we know, 13 -1 (mod 7)

again, for positive integer k, we know, if ab (mod m) then, ak bk (mod m)

hence, 13 -1 (mod 7) gives, 13n (-1)n (mod 7)

again, 6 -1 (mod 7)

hence, adding the two congruences , we get, 13n + 6 (-1)n +1 (mod 7)

if, n is even, then (-1)n = 1 then, (-1)n +1 = 1+1 = 2

then, 13n + 6 2 (mod 7) , then, 7 does not divide 13n + 6 (since, 2 is not congruent to 0 (mod 7) )

now, if n is odd, then (-1)n = -1 then, (-1)n +1 = -1+1 = 0

then, 13n + 6 0 (mod 7) 7 | 13n + 6

hence, for all odd positive integers n, 13n + 6 is divisible by 7

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