Difficulty: Fun

Anamaria and Chen are two friends who toured a magical chocolate factory! They both get enormous amounts of chocolate and they want to share it evenly between them. Anamaria gets 1,574,936 squares of chocolate and Chen gets 3,569,321 squares of chocolate.

Without using a calculator, can you tell if it is possible for them to share the chocolate between them without breaking any squares?

Need a hint?

Those are some big numbers! Rather than doing big calculations, can you think of another way to work it out?

Do you know a special name for a number you can share evenly between two people?

Brainteaser answer

No, the sum of 1,574,936 and 3,569,321 is odd, so it cannot be divided evenly between the two friends.

Divided “evenly” is the key word for this problem! Even numbers, by definition, can be divided by two. Odd numbers don’t divide evenly by two – they leave a remainder of 1.

To know whether the sum of chocolate can be divided evenly, we need to find out whether the sum is even or odd. We can do this by adding the digits in the ones place for both amounts: 6 + 1 = 7, an odd number! This means the sum ends with a 7, which means it is odd (remember you can think of smaller numbers to double check this, like 27 and 107 are both definitely odd).

So, we’ve figured out that the total amount of chocolate is odd, meaning that it does not divide evenly! Chen and Anamaria will have to give the last square to someone else!

Did you know that there are predictable patterns when adding evens and odds?

Two pieces of chocolate are added together, both have an even number of squares. One piece is 2 by 4 and the other is 2 by 2. They make a 2 by 6 piece of chocolate, which also has an even number of squares. The resulting block could be divided into two 1 by 6 pieces.

Adding two even numbers always makes another even number. We can visualise this with two bars of chocolate, both even, both have squares that are all paired up. The resulting block can be divided evenly, right down the middle.

Two pieces of chocolate are added together, both have odd numbers of squares. One piece has 2 by 3 squares with one remainder square sticking out of it and the other is 2 by 1 with a remainder square sticking out of it. These remainders fit together like puzzle pieces to create a 2 by 5 piece of chocolate, which has an even number of squares. The resulting block could be divided into two 1 by 5 pieces.

Adding two odd numbers always makes an even number. Again, we can visualise this with chocolate. Both odd numbers have one remainder, which is the square that sticks out from the rest of the block. But put together, the two remainders pair up, making the sum even. Because 1 plus 1 is 2, it is now possible to split the entire block down the middle into two equal parts.

Two pieces of chocolate are added together, one has an even number of pieces shown as a 2 by 4 block. The other is odd as shown as a 2 by 1 with a remainder square sticking out of it. The resulting block is odd, shown as a 2 by 5 with a remainder square sticking out of it.

Adding an even and an odd number always makes an odd. Check out the chocolate again. In this case, the lonely remainder from the odd number stays as an unpaired remainder. In the problem, 1,574,936 ends with an even digit, so this amount is even. While 3,569,321 ends with an odd digit, making the entire number odd. According to this pattern, the sum these large numbers is odd.

If you don’t believe us, try testing these rules out with smaller numbers and see if the patterns hold true.

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