Division algorithm

• 1 Look at table 1.1. If the same pattern was extended downwards, would it eventually incorporate any positive integer ﹛1, 2, 3, …, n, n +1,…﹜ that we might care to name?

• 2 What is the relation between each number in table 1.1 and the number below it?

• 3 Give a succinct description of the full set of numbers in the column below 0.

• 4 If you choose two numbers from the column below 0 and add them together, where in the table must their sum lie?

• 5 The whole of the array in table 1.1 may be considered as an addition table with the column below 0 down one side and the numbers 1, 2, 3 across the top. Using your brief description of the numbers in the column below 0, devise a comparably succinct description of the full set of numbers in the column below 1, and similarly succinct descriptions of the sets of numbers in the other two columns.

• 6 If two numbers lie in the second column and the lesser is subtracted from the greater, where does the difference lie?

• 7 If two numbers lie in the third column and the lesser is subtracted from the greater, where does the difference lie? Try to prove your result in a general way which would apply to all such pairs.

• 8 If two numbers lie in the fourth column and the lesser is subtracted from the greater, where does the difference lie? Prove it.

• 9 If two numbers are chosen, both from the second column, where does their sum lie? Prove your claim generally.

• 10 If two numbers are chosen, both from the fourth column in table 1.1, where does their sum lie? Prove your claim generally.

• 11 Are there general rules which enable you to fill in the table below for addition of numbers by columns? If only the numbers at the heads of the columns are used in this table, the table that results is an example of an addition table modulo 4. Such a table is denoted by (Z4, +).