You could start with a whole class counting activity:

Start counting together, speaking loudly on the numbers in the
two times table, and quietly on the other numbers. Now split the
class in two. Ask half the class to continue doing the same and ask
the other half to only speak loudly on the numbers in the five
times table.

Which numbers were quiet ?

Which numbers were fairly
loud and which were very
loud ?

Now split the class in three. Two groups to continue as before
and one group to only speak loudly on the numbers in
the three times table.

Can they predict what they will hear?

Which numbers will be quiet?

Which numbers will be fairly loud and which will be very
loud?

Try it.

Class could be split in four and the new group could be asked
to speak loudly on the multiples of four.

When will everyone speak loudly?

Start again and select two numbers which have a
common factor, for example, 4s and 6s.

Ask students to predict which numbers will be spoken
loudly.

Try it.

After this introductory activity, pose a question from the
question generator. Give students a few minutes to think of numbers
that fit one or more of the conditions. Gather some answers and
explanations until the whole group feel
confident suggesting numbers based on statements about divisors and
remainders.

With the same, or a new problem, ask students to work in pairs to find:

a number that fits all the conditions,

then to find all the numbers under 100 that fit them all,

then to write two sentences to explain how they know they have got them all.

With the group together, ask for feedback, and put the answer into the answer box. If you have an interactive whiteboard, it might be appropriate to illustrate the logic with the coloured ball interactivity.

Generate a selection of questions, ask students to pick out particular questions that seem easiest/hardest and work on those. On the board, write "what makes a question like this easy/hard?" and tell students that you'll be collecting suggestions after 15 mins.

If a computer room is available, set students to work at computers in pairs. Students can use the coloured ball interactivity to help them, but emphasise that eventually you would like them to identify the numbers without the aid of the interactivity.

You can print this 10 by 10 number grid so that students can keep a record of their working as they narrow down the possibilities.

Then set the students to play The Remainders Game

Who can reach 100 points in the least number of games?

Ask students to explain any strategies they have generated.

Finally, ask them to have a go at the last question in the problem and emphasise that you will expect themto justify their conclusions.

Which clues are most helpful?

When does a clue provide no new information?

What is the minimum number of divisions needed to identify the
number?

Students, working in pairs, could think of a number themselves
and then give their partners three clues to help them identify
their number, or, as in The
Remainders Game they could allow their partner to choose the
divisors.

Students could also have a go at Ewa's Eggs

Clapping
Times and Music
to My Ears suggest how the introductory activity could be
extended.

Use the coloured ball interactivity and ask students to
predict what will happen.

Flashing
Lights may be an accessible starting problem.