Difficulty: Extreme!
Sam arrives first to her after school Maths Club meeting. She discovers that her teacher has left a basket of Valentine’s Day chocolates. It comes with the instructions to share the chocolate evenly among the six students in the club… but Sam eats a few chocolates before anyone notices!
Next, Sam’s friend Darius arrives, and they work out that if they share the remaining chocolate evenly between the two of them, there is one chocolate left over. When a third student arrives, they find they can split the remaining chocolates evenly three ways with one chocolate left over.
The same thing happens with the fourth, fifth and sixth students. Each time, they try to share the chocolates evenly among the students present but they always have one chocolate left over.
Sam’s teacher left 72 chocolates for all six students to share evenly with no remainders. How many chocolates did Sam eat before Darius arrived?
Need a hint?
A number that can be shared evenly by two people must divide evenly by 2 (it is “divisible” by 2). A number that can be shared evenly by three people must be divisible by three. Can you find a number that is divisible by 2, 3, 4, 5, and 6? Can you find a smaller number that still divisible by these numbers?
Now, if there is always one chocolate left over, the total number of chocolates must be the number you found above plus one.
Brainteaser answer
Sam ate 11 chocolates, leaving left 61 chocolates in the basket when Darius and the other students arrived.
How many chocolates were in the basket after Sam ate her handful? The problem gives lots of clues about this number, so let’s start by figuring it out.
There are a lot of ‘with one left over’ clues, so let’s take one chocolate and put it aside for now. We’ll have to add it back in at the end.
Sharing a number of chocolates evenly among two students means that the number divides evenly by 2 (it is “divisible” by 2). Sharing a number of chocolates evenly among three students means that the number is divisible by three, and so on.
The number of chocolates now in the basket must be divisible by 2, 3, 4, 5, and 6. We can find a possible number by multiplying all of these numbers together to get 720. That’s much more than the 72 pieces that the teacher left! Can we find a smaller number?
Let’s start with 6. It’s the biggest number the chocolates need to be divisible by. Another way of writing 6 is 3 x 2, so if a number is divisible by 6, then it is automatically divisible by 2 and 3 as well.
5 is the next biggest number but it can’t be broken down further like 6. If we multiply 5 x 3 x 2 we get 30, which is divisible by 2, 3, 5 and 6. This only leaves 4.
Multiplying by 4 doesn’t quite work – we get 120 chocolates which is still too much. But 4 can also be written as 2 x 2 and 30 is already divisible by one 2. So we just need to add another one: 2 x 5 x 3 x 2 = 60. Double check that 60 is divisible by 4 (60 ÷ 4 = 15).
Right now, we need to remember the chocolate we put aside at the top and add it on. There are 61 chocolates in the basket. You can try sharing it out to make sure that you always get one left over.
To find out how many Sam ate, it’s a simple subtraction: 72 – 61 = 11 chocolates
There’s also a sneaky second answer. If Sam ate 71 chocolates, there would only be one left in the basket. No matter how you try to share it, there will always be that one left over!
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