Essential supplies
Chocolate bars come in boxes of 5 or boxes of 12. How many boxes do you need to have exactly 2005 chocolate bars?
Age
14 to 16
Challenge level
Problem
'Saturn' chocolate bars are packed in boxes of either 5 or 12.
What is the smallest number of full boxes required to pack exactly 2005 'Saturn' bars?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: $170$
Starting with boxes of 12
${2005}\div{12}=167$ remainder $1$.
So, if $167$ of the larger boxes are used $1$ bar will remain.
If $166$ larger boxes are used there will be $13$ bars left (not a mulitple of $5$).
If $165$ larger boxes are used this will leave $25$ bars, which can be packed into $5$ of the smaller boxes.
Total number of boxes is $165+5=170$.
Starting with boxes of 5
$2005\div5=401$ so could use $401$ small boxes
$12$ boxes of $5$ contain the same number as $5$ boxes of $12$
$401\div12 = 33$ remainder $5$
So $401$ boxes of $5 = 33\times(12$ boxes of $5)$ remainder $5$ boxes of $5$
$ =33\times(5$ boxes of $12)$ remainder $5$ boxes of $5$
$ = 33\times5 + 5 =170$ boxes