# Days and Dates

## Problem

*Days and Dates printable worksheet*

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?**

## Student Solutions

Well done to Chuyi from Loughborough High School, Charkrit from Traill International School, and an anonymous solver, who all explained that as there are seven days in a week, the sequence of Wednesdays will go up in sevens.

Evie from St Anne's School sent us some good thinking about when her birthday will fall:

My birthday is $52$ days away. You divide $52$ by the number of days in a week, so you can find out how many weeks there are. $52 \div 7= 7$r$3$. So my birthday is in 7 weeks and 3 days.

If today is Sunday, we count on 3 days. Therefore, my birthday will be on a Wednesday.

Every year, your birthday shifts forwards one day. This is because $365 \div 7 = 52$r$1$. In $52$ weeks, it will still be a Wednesday, but there is a remainder. So in 2010, my birthday will be on a Thursday.

Of course, a leap year has 366 days so the remainder would be two instead of one, which is why your birthday shifts forward two days every leap year!

Abinhav from Bangkok Patana School sent us a very clear explanation about remainders when we divide by seven:

In $2, 9, 16$ and $23$ days from now, it will be a Wednesday. What other numbers of days from now will be Wednesdays?

Answer: $30, 37, 44, 51$ (keep adding $7$ each time)

Can you generalise what you have noticed?

If the number of days from now is termed $n$, and the remainder upon dividing by $7$ is $r$, then when $n\div 7$ yields $r = 2$, the day will be Wednesday.

From this we can see that if today is a Monday, then the day $n$ days from now will be moved forward $r$ days, starting from Monday. So for example in $24$ days from now, it will be a Thursday since $r$ is $3$. So the day moves forward $3$ days.

Abinhav went on to explain what happens to remainders when you add:

Remainder of numbers ADDED together divided by $7$. If $r_{1}$ is the remainder from the first division, and $r_{2}$ is the remainder from the second division, then the remainder from the total is always $r_{1} + r_{2}$. This occurs since when we add, we simply take the number we had origianlly and add on a supplement. So if originally the number was $9$, giving $r_{1} = 2$; and the second number was $15$, giving $r_{2} = 1$, then $9 + 15 = 24$. This can also be written as $7 + r_{1} + 14 + r_{2}$. So from this we get that $24 = 21 + r_{1} + r_{2}$. Since the number gained from adding multiples of $7$ will always be a multiple of $7$, the final remainder is always $r_{1} + r_{2}$.

Finally, an explanation of what happens to remainders when you multiply:

Remainder when two numbers are multiplied and then divided by $7$. $15 \times 26 = 390$. When $15$ is divided by $7$, remainder is $1$. When $26$ is divided by $7$, r$ = 5$. When $390$ (their product) is divided by $7$, the remainder is $5$.

Hence when we multiply two numbers; upon division the final remainder is also the product of their original remainders. This occurs since when we multiply, we take any number and add those many more lots of that particular number. Say the remainder from the first number is $r_{1}$ and the remainder from the second number is $r_{2}$.

So for $15 \times 26$, we can rewrite this as $$(14 + r_{1})(21 + r_{2})$$ From this we get

$$294 + 14r_{2} + 21r_{1} + r_{1}r_{2}$$

Since the first three terms will always be multiples of 7, the final remainder is always $r_{1}r_{2}$ (in this case, 5).

Of course, in the case where $r_{1}r_{2}$ is greater than 7, the remainder will simply be what would be left if you divided $r_{1}r_{2}$ by 7.

## Teachers' Resources

### 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!

### 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 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.

### 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: