### 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.

### Dozens

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

# Powerful Finale

##### Age 11 to 14 ShortChallenge 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.