Factoring: Coiling Up Clock Numbers

We look at Z/pqZ for distinct primes p and q, showing that it is equivalent to having "numbers" which combine the state of two completely independent clock dials (Z/pZ and Z/qZ). This fact is mathematically written as Z/pqZ = Z/pZ × Z/qZ.

[Numeroscope Applet: displays the clock numbers with variable composite modulus. Can serve as a calculator for this number system, or display additive and multiplicative series, as well as function ranges.]

Lab Activities

Mode Oper. r c Activity
one z 1 Check that the horizontal coordinates of c+z and cz depend only on the horizontal coordinates of c and z. Check the same for vertical coordinates.

one z 1 Check by hand in examples that the rule for how the horizontal and vertical coordinates behave under + and × agrees with those operations in Z/pZ and Z/qZ, while the numbers themselves work exactly like those in the displayed clock dial for Z/pqZ.

one z 1 0 Notice that because 0×(anything) = 0, if you set c to zero, multiplying by anything gets you nowhere. Now pick a c which is not zero, but is outside of the shaded region. what happens when you multiply it by different numbers? Which numbers can you get as products with c?

one z 1 Check that in each instance the numbers at a at (1,0) and b at (0,1) have a special property: their squares are themselves, while their product is zero. By hand, verify that based on these properties, (ax+by) + (ax'+by') = a(x+x') + b(y+y') and (ax+by)(ax'+by') = (axx'+byy'), so that we get an algebraic verification of the fact that addition and multiplication work coordinatewise.

sweep/fill r + 0 Notice how addition simply proceeds in straight lines, except for the "wraparound" effect.

all r + 0 Which additive series (of multiples of z) fill up the whole number system? What are all the different possible results of filling in all multiples? How does this correspond to the previous question?

fill/sweep r × 1 Notice how the effect of multiplication does not have any geometrically obvious interpretation -- at least in this view.

all r × 1 What are all the different possible results of filling in all powers of a number? How do the results compare with those for clock numbers modulo a prime? Specifically, try the following combinations, and compare:
p 11 13 13 q 7 5 7

*Mode*	*Oper.*	r	c	*Activity*
one z		1		Check that the horizontal coordinates of c+z and cz depend only on the horizontal coordinates of c and z. Check the same for vertical coordinates.
one z		1		Check by hand in examples that the rule for how the horizontal and vertical coordinates behave under + and × agrees with those operations in Z/pZ and Z/qZ, while the numbers themselves work exactly like those in the displayed clock dial for Z/pqZ.
one z		1	0	Notice that because 0×(anything) = 0, if you set c to zero, multiplying by anything gets you nowhere. Now pick a c which is not zero, but is outside of the shaded region. what happens when you multiply it by different numbers? Which numbers can you get as products with c?
one z		1		Check that in each instance the numbers at a at (1,0) and b at (0,1) have a special property: their squares are themselves, while their product is zero. By hand, verify that based on these properties, (ax+by) + (ax'+by') = a(x+x') + b(y+y') and (ax+by)(ax'+by') = (axx'+byy'), so that we get an algebraic verification of the fact that addition and multiplication work coordinatewise.
sweep/fill r	+		0	Notice how addition simply proceeds in straight lines, except for the "wraparound" effect.
all r	+		0	Which additive series (of multiples of z) fill up the whole number system? What are all the different possible results of filling in all multiples? How does this correspond to the previous question?
fill/sweep r	×		1	Notice how the effect of multiplication does not have any geometrically obvious interpretation -- at least in this view.
all r	×		1	What are all the different possible results of filling in all powers of a number? How do the results compare with those for clock numbers modulo a prime? Specifically, try the following combinations, and compare: p 11 13 13 q 7 5 7

Discussion Topics

What is the significance of the shaded region with respect to multiplication?

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