### 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

##### Stage: 4 Challenge Level:

If today is Monday we know that in 702 days' time (that is in 100 weeks and 2 days' time) it will be Wednesday. This is an example of "clock" or "modular" arithmetic.

What day will it be in 15 days? 26 days? 234 days?

In 2, 9, 16 and 23 days from now, it will be a Wednesday.

What other numbers of days from now will be Wednesdays?
Can you generalise what you have noticed?

Choose a pair of numbers and find the remainders when you divide by 7.
Then find the remainder when you divide the total by 7. For example:

 $15 \div 7 = 2$ remainder $1$ $15 + 26 = 41$ $26 \div 7 = 3$ remainder $5$ $41 \div 7 = 5$ remainder $6$

Choose some more pairs of numbers.
Is there a relationship between the remainders when you divide each by 7, and the remainder when you divide their total by 7?

Now find the remainder when you divide the product of 15 and 26 by 7. What happens?

Choose some more pairs of numbers.
Is there a relationship between the remainders when you divide each by 7, and the remainder when you divide their product by 7?

What about when you divide by numbers other than 7?
Can you explain what you've noticed?

How could you use these ideas to work out on which day of the week your birthday will be next year?