2015 AMC 12B Problems/Problem 18
Contents
Problem
For every composite positive integer , define to be the sum of the factors in the prime factorization of . For example, because the prime factorization of is , and . What is the range of the function , ?
Solution
Solution 1
This problem becomes simple once we recognize that the domain of the function is . By evaluating to be , we can see that is incorrect. Evaluating to be , we see that both and are incorrect. Since our domain consists of composite numbers, which, by definition, are a product of at least two positive primes, the minimum value of is , so is incorrect. That leaves us with .
Solution 2
Think backwards. The range is the same as the numbers that can be expressed as the sum of two or more prime positive integers.
The lowest number we can get is . For any number greater than 4, we can get to it by adding some amount of 2's and then possibly a 3 if that number is odd. For example, 23 can be obtained by adding 2 ten times and adding a 3; this corresponds to the argument . Thus our answer is .
Solution 3
Notice that by the Chicken McNugget Theorem, given s and s, we can make any number larger than We also exclude and since they correspond to primes. All other numbers can be written as which will equal. Thus the answer is
Solution 4 (Elimination)
Note that eliminates and and eliminates . We can easily see that cannot be less that by testing and . So, the answer is clearly .
See Also
2015 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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