Find the number of positive integers that satisfy the condition above.
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We note that 7 ∣ 5 n + 1 ⟹ 5 n ≡ − 1 (mod 7) . Consider 5 n ≡ ( 7 − 2 ) n ≡ ( − 2 ) n (mod 7) . For the first few n , we have ( − 2 ) n = − 2 , 4 , − 1 , 2 , − 4 , 1 , − 2 , . . . . We note that ( − 2 ) n has a period of 6 and ( − 2 ) n = − 1 , when n m o d 6 = 3 . For n ≤ 1 0 0 0 , the acceptable n = 3 , 9 , 1 5 , . . . 9 9 9 , a total of 1 6 7 of them.