The number is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
- A)
- B)
- C)
- D)
- E)
Answer
B
Key insight
Twenty two-digit numbers total at most 20·99 = 1980 < 2024, while twenty 99s plus one 44 reach 2024 exactly, so 21 is the least.
Solution
Each two-digit number is at most , so of them add to at most , which is less than . Twenty (or fewer) numbers cannot work.
Twenty-one can: , and is a two-digit number. So twenty s and one do the job.
The answer is .
Why this works
"Least number of summands" is a two-part argument: an upper bound on what summands can reach (use the largest allowed value) and an explicit construction for the first that is not ruled out. Using as many maximal pieces as possible is the natural construction.
The trap
Dividing 2024 by 100 and answering 20, forgetting that the largest two-digit number is 99 and twenty of them fall short.
Common mistakes
- Dividing 2024 by 100 and answering 20, forgetting that the largest two-digit number is 99 and twenty of them fall short.
- Finding a valid decomposition with or more terms and not checking whether fewer suffice.
Techniques
Bound the quantity above/below or estimate to pin it down · Consider the largest/smallest element or boundary case