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N000ughty Thoughts

How many noughts are at the end of these giant numbers?

DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

Essential Supplies

Age 14 to 16 Short
Challenge Level

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

This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.