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?**

Can you explain what you've noticed?