SOLUTION: 2. If a teacher has less than 100 pieces of candy and puts them into groups of 2, 3, and 4 groups there is always one left over, if she puts the pieces of candy into groups of 5, t

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Question 922661: 2. If a teacher has less than 100 pieces of candy and puts them into groups of 2, 3, and 4 groups there is always one left over, if she puts the pieces of candy into groups of 5, there are none left over. How many pieces of candy were there?
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2. If a teacher has less than 100 pieces of candy and puts them into groups of 2, 3, and 4 groups there is always one left over, if she puts the pieces of candy into groups of 5, there are none left over. How many pieces of candy were there?
Answer:
Let's call the unknown number of candies C.
Since "if she puts the pieces of candy into groups of 5, there are none left over", the number of candies must be a multiple of 5. So, the possible solutions are 5, 10, ... 95 (since there are less than 100).
Now consider the fact that dividing C by 4 leaves a remainder of one. Which multiples of 5 - the possible solutions - when divided by 4 leave a remainder of 1. Not the ones that end in 0, like 10, 20 etc., because to produce a remainder of one the number one below the 10, 20, etc. would have to be a multiple of 4. Since all of the numbers one below 10, 20, etc. end in 9, like 9, 19, 29 etc. they are all odd numbers, which cannot be multiples of 4. So we can eliminate all the multiples of 10 as the solution.
This leaves the numbers ending in 5. If we look at these, the numbers that are one less than those numbers that are multiples of 4 are: 4, 24, 44, 64 and 84. So the possible answers for C are now down to 5, 25, 45, 65, and 85.
Let us turn to the fact that dividing C by 3 gives a remainder of 1. Looking at the possible answers for see, we see two of them give a remainder of 1 when divided by 3: 25 and 85.
Let's see if dividing by 2 (and getting a remainder of 1) helps us decide which of 25 and 85 is the answer. It turns out that dividing these numbers by 2 both give a remainder of 1.
So we know that the teacher has either 25 or 85 candies. The problem does not give us enough information to decide which. But if we are a student in the class, we are certainly hoping for 85!