Kyra has a rock collection. When she puts her rocks into 2 equal piles, there are no rocks left over. When she puts her rocks into 3 equal piles, there are still no rocks left over. When she puts her rocks into 4 equal piles, there are still no rocks left over. How many rocks could Kyra have
The Correct Answer and Explanation is:
Kyra could have 12 rocks.
Explanation
We are told that Kyra’s rocks can be divided evenly into 2, 3, and 4 equal piles with no rocks left over. This means the number of rocks must be divisible by 2, 3, and 4. In mathematics, a number that is divisible by several other numbers is called a common multiple of those numbers. The least such number is called the Least Common Multiple (LCM).
To find the smallest possible number of rocks Kyra could have, we calculate the LCM of 2, 3, and 4.
Step 1: Prime Factorization
- 2 is a prime number → 2
- 3 is a prime number → 3
- 4 = 2 × 2 = 2²
Step 2: Choose the highest powers of each prime
- From 2: 2² (from 4)
- From 3: 3¹
Step 3: Multiply them
LCM = 2² × 3 = 4 × 3 = 12
So, the least number of rocks Kyra could have is 12.
Let’s verify:
- 12 ÷ 2 = 6 → no remainder ✅
- 12 ÷ 3 = 4 → no remainder ✅
- 12 ÷ 4 = 3 → no remainder ✅
Therefore, 12 rocks meet all the conditions.
However, the problem says “how many rocks could Kyra have?”, not “what is the smallest number?” This means any multiple of 12 would also work (like 24, 36, 48, etc.). But since no specific upper limit is given, the correct answer is the smallest possible, which is 12.
In real-world contexts, problems like this teach children about divisibility, multiples, and least common multiples. It’s a great way to apply basic number theory to a scenario that feels concrete and relatable, such as organizing a rock collection.
