1986 AIME Problems/Problem 5
What is that largest positive integer for which is divisible by ?
Video Solution by OmegaLearn
If , . Using the Euclidean algorithm, we have , so must divide . The greatest integer for which divides is ; we can double-check manually and we find that indeed .
Solution 2 (Simple)
Let , then . Then Therefore, must be divisible by , which is largest when and
In a similar manner, we can apply synthetic division. We are looking for . Again, must be a factor of .
The key to this problem is to realize that for all . Since we are asked to find the maximum possible such that , we have: . This is because of the property that states that if and , then . Since, the largest factor of 900 is itself we have:
Solution 5 (Easy Modular Arithmetic)
Notice that . Therefore
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