Final answer:
The least number of pennies that can be arranged in piles containing exactly 9, 10, 11, and 12 pennies is found by identifying the least common multiple of these numbers, which is 360. However, none of the options provided correctly represents this number. Therefore, a mistake may have occurred in the question's options.
Step-by-step explanation:
The student is looking for the least number of pennies that can be arranged into piles containing exactly 9, 10, 11, and 12 pennies. To find this number, we need to find the smallest common multiple of 9, 10, 11, and 12. Here is the step-by-step explanation:
- List multiples of each number until we find common ones: Multiples of 9 (9, 18, 27, 36, 45, 54, ...), 10 (10, 20, 30, 40, 50, 60, ...), 11 (11, 22, 33, 44, 55, 66, ...), and 12 (12, 24, 36, 48, 60, 72, ...).
- Identify the least common multiple: The least multiple that appears in all lists is 360.
- Compare with the options: We notice that 360 pennies are not among the given options, which implies we need to find the next smallest multiple. Continuing our search would reveal that the next smallest common multiple is 396, which is also absent from the options.
However, upon closer inspection, it seems there may have been a misunderstanding in the options provided, as none of them are common multiples of 9, 10, 11, and 12. The smallest number of pennies that fulfill the criteria is not listed. Therefore, none of the options (a) 45, (b) 90, (c) 99, (d) 108 are correct. The student should double-check the question or consider that a mistake has occurred in the given options.