Parallelogram 27 • Level 1 • 7 Mar 2024Prime time Level 1

Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

a portmanteau word combining parallel and telegram. A message sent each week by the Parallel Project to bright young mathematicians.

Tackle each Parallelogram in one go. Don’t get distracted.
Finish by midnight on Sunday if your whole class is doing parallelograms.
Your score & answer sheet will appear immediately after you hit SUBMIT.
Don’t worry if you score less than 50%, because it means you will learn something new when you check the solutions.

1. Prime Reciprocals

Primes, famously, can only be divided equally by 1 and themselves - but what happens if you divide 1 by a prime?

Watch this video where Matt Parker spends a day in the Royal Society archives getting to the bottom of this mystery.

(If you have problems watching the video, right click to open it in a new window)

1 mark

1.1 What happens (usually) when you divide 1 by a prime number?

The result is a whole number
It creates an infinitely looping decimal
It creates a decimal which is infinitely long and never repeats
The universe implodes
(Not answered)

1 mark

1.2 There are two prime numbers which are exceptions to this rule, what are they?

1 and 11
47 and 51
2 and 5
47 and 53
(Not answered)

1 mark

1.3 How long is the repeating part of 113?

5 digits
6 digits
7 digits
It doesn’t repeat
(Not answered)

When 1 is divided by 13 the answer is 0.076923076923076923...

You can see the string 076923 repeats and so the period of 113 is 6 digits.

2 marks

1.4 We know that 17 gives a repeating decimal, but what about 27?

Try to find it without a calculator.

0.142857
0.428571
0.285714
0.857142
(Not answered)

With prime reciprocals the different numerators cycle through a pattern on the denominator.

Look:

17 = 0.142857
37 = 0.428571
27 = 0.285714
67 = 0.857142
47 = 0.571428
57 = 0.714285

2. In your prime

This is only the start of the fun properties of primes.

For the next two questions, it helps to know the prime numbers up to 100.

2 marks

2.1 Add up the first 9 primes to find a satisfying result!

Its a square number
Its also a prime
Its a single digit recurring
Its a palindromic number
(Not answered)

The sum of the first 9 primes is:

2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 = 100

Look at this satisfying visual proof:

3 marks

2.2 Square numbers like 100 can even be made by adding just two primes e.g. 97 + 3 = 100.

How many different pairs of prime add for a total of 100?

2
4
6
8
(Not answered)

Show Hint (–1 mark)

–1 mark

It may help to look at a list of all the prime numbers up to 100.

Then try to pair them up to make 100.

Here are two examples:

3 + 97 = 100
11 + 89 = 100

The question is asking how many prime pairs can make 100.

Here are all of the possibilities for adding exactly two primes to total 100:

97 + 3
89 + 11
83 + 17
71 + 29
59 + 41
53 + 47

In fact, all square numbers up to 100 can be made by adding two primes together - try a few yourself!

Before you hit the SUBMIT button, here are some quick reminders:

You will receive your score immediately, and collect your reward points.
You might earn a new badge... if not, then maybe next week.
Make sure you go through the solution sheet – it is massively important.
A score of less than 50% is ok – it means you can learn lots from your mistakes.
The next Parallelogram is next week, at 3pm on Thursday.
Finally, if you missed any earlier Parallelograms, make sure you go back and complete them. You can still earn reward points and badges by completing missed Parallelograms.

Cheerio, Simon and Ayliean.