1998 AIME Problems/Problem 1
Problem
For how many values of is the least common multiple of the positive integers , , and ?
Solution 1
It is evident that has only 2s and 3s in its prime factorization, or .
The LCM of any numbers an be found by writing out their factorizations and taking the greatest power for each factor. . Therefore , and . Since , there are values of .
Solution 2
We want the number of such that .
Using properties, this is , or .
At this point, we realize that , as any other prime factors would be included in the .
Also, (or the power of in the wouldn't be ) and (or the power of in the would be and not ).
Therefore, can be any integer from to , for a total of values of and values of .
Video Solution by OmegaLearn
https://youtu.be/HISL2-N5NVg?t=2899
~ pi_is_3.14
See also
1998 AIME (Problems • Answer Key • Resources) | ||
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