### Why do this problem?

This problem is one which
uses many properties of numbers such as factors, multiples, and
primes. It gives learners a chance to invent their own variations
on the theme of finding a number that someone else has in mind.

### Possible approach

You could start by
playing "Twenty questions" with the whole group with such numbers
as $56$ ($7 \times 8$), $71$ (prime number), $78$ ($13 \times 6$),
$111$ ($3 \times 37$) and $322$ ($161 \times 2$) to bring up
multiples, factors, prime and palendromic numbers. You could also
discuss the various methods used to test whether a number is
divisible by $2, 3, 9, 11$ etc. It would also be useful to talk
about the best questions to ask first and those which are best left
until later.

Invite learners to work
in pairs on the actual problem - having it printed out would be
helpful - so that they are able to talk through their ideas with a
partner. Warn them that the plenary will focus on how they found the solution, not
just the answer itself.

After a suitable length
of time, bring the group together to discuss the various clues that
were most helpful in reaching it. Did they use any tests for
divisibility? Did they use their knowledge of prime numbers? At
what point were they fairly certain of Tom's number?

### Key questions

What do you know about
Tom's number?

Can you estimate what it
might be and then check?

### Possible extension

Learners could choose numbers that they consider would be good to
use in their own version of this game and then suggest some
suitable questions to ask about them.

### Possible support

Some learners might benefit from making a list of possible numbers,
or criteria, as they read through the clues.