There is a shop with $5$ items of the following costs:
Item A = $\$300$
Item B = $\$750$
Item C = $\$1,500$
Item D = $\$3,000$
Item E = $\$6,000$
You currently have $\$200$ worth of credit for the shop, but you have unlimited funding of money. If you may only spend in increments of $\$300$, $\$750$, $\$1,575$, $\$3,300$, $\$6,900$, and $\$12,000$, what is the least amount of money you can spend purchasing any combination of these items so that you have exactly $\$0$ worth of credit remaining for the shop? If no such solution exists, explain why.
So basically I have to minimize $$300a + 750b + 1,500c + 3,000d + 6,000e = 200 + 300z + 750y + 1,575x + 3,300w + 6,900v + 12,000u$$ for domain in the integers greater than or equal to zero.
Because there's so many possibilities due to the lack of restrictions of the variables other than integers greater than or equal to zero, I'm not sure where to start other than brute-forcing different possible combinations.
No matter what you buy, your total bill will be a multiple of $\$75$. All of your possible spending amounts are multiples of $\$75$. The only thing in sight that's not a multiple of $\$75$ is the $\$200$ store credit you're trying to get rid of. Your task is impossible. You can't add $200$ to a multiple of $75$ and arrive at a multiple of $75$.
It's like trying to exactly use a $\$4$ gift card when all you have are $\$5$ bills, and every price is in $\$5$ increments. Can't be done.