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What is the units digit for the number 123^(456) ?

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What is the largest number which, when divided into 1905, 2587, 3951, 7020 and 8725 in turn, leaves the same remainder each time?

Galley Division

Stage: 4 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

Why do this problem :


This problem draws attention to the way digits are arranged in a written calculation. The traditional representation for basic computation can be taken for granted. This problem allows the display choices made to become conspicuous and appreciated.

Possible approach :


Ask the group to do the first calculation on a calculator to get 'an answer' and then ask questions about what that decimal display means.

Draw attention to the fact that people have been successfully calculating results of value for centuries, long before electronic calculation, and ask what calculations might have been important and why.

Ask how these calculations might have been achieved and in particular why some of our standard non-calculator algorithms work (include long multiplication, and long division, plus finding prime factors if that is within the students' experience).

Draw the students into a similar discussion about the Galley method for arranging and holding digits during a calculation.

Ideally this should be group work with lots of talk : noticing possible positioning of digits, reasoning to support or challenge the validity of those conjectures, and gradually building up a complete understanding of the whole method. This can then be practised with new numbers, rehearsing the reasoning in the process.

Key questions :


  • What does 'division' mean ?
  • How would you work out 65284 divided by 594 ?
  • Without a calculator ?
  • How do we know that's an 'OK method' ?
  • Any thoughts on how the Galley way of setting out the calculation works ?

Possible extension :

Composite Notions
Unusual Long Division - Square Roots Before Calculators

Possible support :

Multiplying with Lines