Mock AIME 3 Pre 2005 Problems/Problem 12

Problem

Determine the number of integers $n$ such that $1 \le n \le 1000$ and $n^{12} - 1$ is divisible by $73$ .

Solution

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We see a pattern when we look at the numbers that do fulfull this property. The first number is $1$ . Then $3, 8, 9, 24, 27, ....$ . This follows a pattern. The first number being $1$ , and the rest being the previous: $+2, +5, +1, +15, +3, +19, +3, +15, +1, +5, +2$ . This sequence then repeats itself. We hence find that there are a total of $11*15 - 1$ or $\boxed{164}$ numbers that satisfy the inequality.

Mock AIME 3 Pre 2005 (Problems, Source)
Preceded by Problem 11	Followed by Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

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Mock AIME 3 Pre 2005 Problems/Problem 12

Problem

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