Tom's Number

'Tom's Number' printed from https://nrich.maths.org/

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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.