Parallelogram 21 Level 1 25 Jan 2024A perfect number

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Noun: Parallelogram Pronunciation: /ˌparəˈlɛləɡram/

  1. 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. 8128 and Perfect Numbers

Yes, all numbers are wonderful - but how many of them are perfect?

So far, only 47.

Watch this video explaining some quirks of the rare and elusive perfect number.

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

1 mark

1.1 What makes a number perfect?

  • All of its digits are even
  • It is even and prime
  • If you add up the proper divisors you get the number itself
  • All numbers are perfect!
  • (Not answered)
1 mark

1.2 How many odd perfect numbers are there?

  • Infinitely many
  • Only one
  • None
  • We don’t know yet
  • (Not answered)

Although all of the 47 perfect numbers we have found so far are even, we don’t yet know if there are any odd ones - or even how many perfect numbers there are!

2 marks

1.3 What is the smallest number with exactly five divisors (including the number itself)?

  • 5
  • 12
  • 16
  • 36
  • (Not answered)
2 marks

1.4 Four only has three factors: 1, 2, and 4.

Which other numbers have an odd amount of factors?

  • All even numbers
  • All square numbers
  • All powers of two
  • None, 4 is the only example of a number with an odd amount of factors
  • (Not answered)

2. What number am I thinking of?

2 marks

2.1 Ayliean thinks of a number, divides it by 5, adds 20 to it and ends up with the number she first thought of.

What was the number?

Correct Solution: 25

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.