A teacher wants to give each sudent in his class some candy. He wants everyone to get an equal number of pieces. The teacher has fewer than 85 pieces of candy. If he arranges the pieces in froups of 2, 3 or 4 for each student, he wil have 1 piece left over. if he uses groups of 5, he will have none left over. How many pieces of candy does he have?
I am so lost, it's not even funny. what could I do other than guess and check?
Thanks!
The number must be a multiple of 5; any number that ends in 0 or 5 is a multiple of 5. The number is not a multiple of 2, 3, or 4.
That should get you started.
if guessed and checked, and I found it to be 25.
5 x 5 = 25
4 x 6 = 24
3 x 8 = 24
2x 12 = 24
so, do you have to guess and check, or is there another way to find that?
To solve this problem, we can use a systematic approach rather than guessing and checking. We will use the concept of the least common multiple (LCM) to find the solution.
Let's analyze the given information:
1. The candy needs to be divided into equal groups for each student.
2. The candy can be arranged in groups of 2, 3, 4, or 5, with leftovers of 1 or 0 depending on the group size.
We can begin by finding the LCM of 2, 3, 4, and 5. The LCM is the smallest multiple that is divisible by all the given numbers.
1. Start by finding the LCM of 2 and 3. The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, ... and the multiples of 3 are 3, 6, 9, 12, 15, 18, ...
The least common multiple of 2 and 3 is 6.
2. Now, find the LCM of 6 and 4. The multiples of 6 are 6, 12, 18, 24, ... and the multiples of 4 are 4, 8, 12, 16, 20, ...
The least common multiple of 6 and 4 is 12.
3. Lastly, find the LCM of 12 and 5. The multiples of 12 are 12, 24, 36, ... and the multiples of 5 are 5, 10, 15, 20, 25, 30, ...
The least common multiple of 12 and 5 is 60.
Therefore, the teacher must have at least 60 pieces of candy to distribute them equally among the students.
Now, let's check if this solution satisfies the given conditions:
- Dividing 60 by 2 leaves a remainder of 0.
- Dividing 60 by 3 leaves a remainder of 0.
- Dividing 60 by 4 leaves a remainder of 0.
- Dividing 60 by 5 leaves a remainder of 0.
Hence, the answer is 60 pieces of candy.
By using the concept of the least common multiple, we were able to find a systematic solution rather than just guessing and checking.