### More Mods

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

### Mod 3

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

### Novemberish

a) A four digit number (in base 10) aabb is a perfect square. Discuss ways of systematically finding this number. (b) Prove that 11^{10}-1 is divisible by 100.

# Days and Dates

### Why do this problem?

This problem gives an insight into modular arithmetic without worrying too much about notation, by looking at the concept of remainders. It gives students the opportunity to share ideas, listen to each other justify their assertions, and come up with convincing arguments and proofs using simple algebra.

### Possible approach

This printable worksheet may be useful: Days and Dates.

Start by making sure everyone is convinced that 702 days after a Monday will be a Wednesday, by thinking about whole numbers of weeks and days left over. Students can then work out what day it will be in 15 days, 26 days, 234 days. Make sure everyone understands that for the purposes of this problem we are always counting from Monday!

Pose the question "If today is Monday, how many days from now is Wednesday?" Ask the students to give you as many answers as they can. (Does anyone suggest a negative number of days?) Ask them to come up with a generalisation (possibly algebraic) for any Wednesday. When discussing their generalisations, focus on considering the number of days in a whole number of weeks with 2 extra, rather than simply extending the pattern in the sequence 2, 9, 16, 23...

Now they are ready to investigate the effects of adding or multiplying numbers on the remainder when we divide by 7. It may be worthwhile to do an example as a group:
$15 \div 7 = 2$ remainder $1$
$26 \div 7 = 3$ remainder $5$
$15 + 26 = 41$
$41 \div 7 = 5$ remainder $6$

Then give the students time to try a few examples of their own and write down what they notice. Make sure they can explain what happens when the remainders of each number add up to more than 7.
They can justify what they have noticed, possibly by using algebra or by giving a convincing argument based on whole numbers of weeks and days left over.

### Key questions

What will numbers have in common if they take us to a particular day of the week?
If the first day of this month was ... what can we say about the first day of next month, and why?

### Possible extension

Investigate patterns when dividing by numbers other than 7. Does the same thing always happen? Students could be introduced to the language and notation of modular arithmetic; if the remainder is 2 when we divide 23 by 7, we write:
$23 \equiv 2$ mod $7$
and say "23 is congruent to 2 mod 7"
Further reading on modular arithmetic can be found here.

### Possible support

Students who are struggling with finding the remainder when dividing by 7 could investigate division by 5 instead; this could be a model for the working week ignoring Saturday and Sunday.