Grandma's cake
What is the smallest number of pieces grandma should cut her cake into to guarantee each grandchild gets the same amount of cake and none is left over?
Age
11 to 14
Challenge level
Problem
Grandma bakes a cake for her grandchildren who are going to visit her.
She has forgotten whether $3$, $5$ or all $6$ of her grandchildren will visit.
She wants all of the cake to be eaten by her grandchildren and would like each grandchild to get the same amount of cake.
To prepare for all three possibilities, what is the smallest number of pieces into which she should cut the cake?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 30, if the slices are all the same size
10, if the slices can be differently sized
Grandma needs to have a number of pieces of cake that is divisible by $3$, $5$ and $6$. The smallest such number will be the Lowest Common Multiple of these numbers.
$3$ and $5$ are prime numbers, whilst $6$ factorises as $2 \times 3$. The lowest common multiple will therefore have prime factorisation $2 \times 3 \times 5$.
This means the lowest common multiple will be $30$, so Grandma should cut her cake into $30$ pieces.
Michael Hatch sent us two alternative solutions:
As the wording of the question states each child must have the same AMOUNT of cake, not the same number of slices, and there is also no condition that the pieces need to be cut into the same sized slices, this creates a different solution.
Only 10 slices of cake are needed: 5 slices of $60^\circ$ and 5 slices of $12^\circ$.
3 grandchildren: 2 have 2 slices of $60^\circ$, 1 has 1 slice of $60^\circ$ and 5 slices of $12^\circ$.
5 grandchildren: each has 1 slice of $60^\circ$ and 1 slice of $12^\circ$.
6 grandchildren: 5 have 1 slice of $60^\circ$ and 1 has 5 slices of $12^\circ$.
Fred also suggested a way of cutting the cake into just 10 slices:
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