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Counting Factors

Is there an efficient way to work out how many factors a large number has?

Big Powers

Three people chose this as a favourite problem. It is the sort of problem that needs thinking time - but once the connection is made it gives access to many similar ideas.


Can you select the missing digit(s) to find the largest multiple?

Powerful Finale

Age 11 to 14 Short
Challenge Level

Answer: $7$

To find the next power of $3$, multiply by $3$
To find the last digit of the next power of $3$, multiply the last digit by $3$

$3^1 = 3$                                       $3^5 = ...3$, since $1\times3=3$
$3^2=9$                                       $3^6=...9$, since $3\times3=9$
$3^3=...7$                                    $3^7=...7$                                   $3^{2011}= ...7$
$3^4=...1$, since $7 \times 3 = 21$         $3^8=...1$                                   $3^{2012}= ...1$

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