2023 AMC 12B Problems/Problem 16
Solution
This problem asks to find largest 
that cannot be written as
![\[ 6 a + 10 b + 15 c = x, \hspace{1cm} (1) \]](http://latex.artofproblemsolving.com/4/5/9/4599a23c5484a8b1490fa8d5b7849f3f2c7a44ef.png)
where 
.
Denote by 
the remainder of divided by 2.
Modulo 2 on Equation (1), we get
By using modulus 
on the equation above, we get
.
Following from Chicken MuNugget's theorem, we have that any number that is no less than 
can be expressed in the form of 
with 
.
Therefore, all even numbers that are at least equal to 
can be written in the form of Equation (1) with .
All odd numbers that are at least equal to 
can be written in the form of Equation (1) with .
The above two cases jointly imply that all numbers that are at least 30 can be written in the form of Equation (1) with .
Next, we need to prove that 29 cannot be written in the form of Equation (1) with .
Because 29 is odd, we must have 
.
Because , we must have 
.
Plugging this into Equation (1), we get 
.
However, this equation does not have non-negative integer solutions.
All analysis above jointly imply that the largest that has no non-negative integer solution to Equation (1) is 29.
Therefore, the answer is 
.
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
