2021 Fall AMC 12A Problems/Problem 8
Problem
Let be the least common multiple of all the integers through inclusive. Let be the least common multiple of and What is the value of
Solution
By the definition of least common mutiple, we take the greatest powers of the prime numbers of the prime factorization of all the numbers, that we are taking the of. In this case, Now, using the same logic, we find that because we have an extra power of and an extra power of Thus,
~NH14
Solution 2
We know that between 32 and 40, there is one prime number, namely . This was not included in the previous , so we know that the value of will include a new 37. Then, let’s consider each individual number. has a new power of 2 now included in , so we add that to our list. , which was included in the previous . Similarly, we can go through the numbers 32 to 40 and find that the only new powers of numbers added are , which gives
~MathCosine
Video Solution by TheBeautyofMath
https://youtu.be/wlDlByKI7A8?t=410
~IceMatrix
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 7 |
Followed by Problem 9 |
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All AMC 12 Problems and Solutions |