Paul has a package of bubble gum. He can devide his gum equally between two people with no pieces left over. He can also devide his gum equally between three people with no pieces left over.
How many pieces of gum could the package contain? Explain how you found your answer.
6, 12, and 18 are divisible by 2 and 3.
To find the number of pieces of gum in the package, we need to determine the common multiple of 2 and 3 since Paul can divide his gum equally between two and three people without any pieces left over.
First, let's find the multiples of 2:
2, 4, 6, 8, 10, 12, ...
Now, let's find the multiples of 3:
3, 6, 9, 12, 15, 18, ...
From these lists, we can see that the first common multiple of 2 and 3 is 6. Since Paul can divide the gum equally between two or three people, the package of gum could contain 6 pieces.
To double-check, we can divide 6 pieces of gum equally between two people: 6 ÷ 2 = 3 pieces per person. We can also divide 6 pieces of gum equally between three people: 6 ÷ 3 = 2 pieces per person. In both cases, there are no pieces left over.
To find the number of pieces of gum in the package, we need to determine the common factors that satisfy the given conditions.
Let's assume that the total number of pieces of gum in the package is "x".
Since Paul can divide the gum equally between two people with no pieces left over, this means that x must be divisible by 2.
Similarly, since he can divide the gum equally between three people with no pieces left over, this means that x must also be divisible by 3.
To find the values of x that satisfy both conditions, we need to find the least common multiple (LCM) of 2 and 3.
The prime factorization of 2 is 2, and the prime factorization of 3 is 3.
To find the LCM, we take the highest power of each prime factor. In this case, both 2 and 3 have a power of 1.
Therefore, the LCM of 2 and 3 is simply 2 * 3 = 6.
So, the package of gum could contain 6 pieces.