2015 AMC 12B Problems/Problem 18
Contents
[hide]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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