Draft: this lesson was written with AI help and has not yet been read by a human editor.
1 What your class asks
This lesson asks you to understand the divisibility rules and use them with 2, 3, 4, 5, 6, 8, 9 and 10, and to write composite numbers as products of prime factors, for example with factor trees. The museum offers one record about numbers and their divisors; the rules and the trees are for your class. 1
Curriculum expectations, with their official codes
- B2.2 understand the divisibility rules and use them to determine whether numbers are divisible by 2, 3, 4, 5, 6, 8, 9, and 10
- B2.6 represent composite numbers as a product of their prime factors, including through the use of factor trees
2 The lesson
How can you tell, without dividing, what divides a number?
Look at the expectations: they ask for rules, not for long division. Test each rule on numbers you can check, and use a factor tree to see which prime factors you cannot split further.
Go deeper
Write the divisors of 220 that are smaller than 220: 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110. Their sum is 284. Now do the same for 284, whose divisors are 1, 2, 4, 71 and 142: they add up to 220 2. Two different numbers that lead to each other like this are called amicable 3. Kamal al-Din al-Farisi wrote the Memorandum for friends on amicability, which gives a new approach to proving an older rule 4. The record calls this exercise modern arithmetic, not an instruction from a manuscript.
Why they did it
The records read here preserve no sentence in which al-Farisi says why he wrote about amicable numbers. They show what the work did: MacTutor treats it as more than a minor change to the older proof of Thabit ibn Qurra, a new approach employing factorization and combinatorial ideas 5. They also show that the same man worked on light: the biography itself distinguishes these two major contributions, the numbers and the water-sphere rainbow model 6.
Often misunderstood
Al-Farisi did not invent the example of 220 and 284, or Thabit ibn Qurra's older rule 3 4. MacTutor also reports Hogendijk's argument that Thabit may already have known the larger pair, 17296 and 18416, so the record awards al-Farisi no first discovery 7. And calling the numbers friends is a verbal analogy, not a mechanism governing their behavior 3.
Think about it
The number 6 has smaller divisors 1, 2 and 3, and they add up to 6. Why does the record not call 6 an amicable pair with itself, and what must the second number do for two numbers to be amicable?
Into the museum
- al-Farisi's factorization-based amicability method: the divisor sums for 220 and 284, and a pair where the rule fails.
- Memorandum for friends on amicability: the treatise and what is not known about its date.
- Kamal al-Din al-Farisi: the mathematician, and his work on optics.
Try this
Read the record on al-Farisi's method for amicable numbers. List the divisors of 220 and add the ones that are smaller than 220, then try the same with 284.
3 Check yourself
Nothing here is marked or sent anywhere. It is for you.
In your own words: what does Al-Farisi's factorization-based amicability method have to do with this lesson?
Compare with the card
Two numbers that give each other. A number can yield another when its proper divisors are added, and the other's divisors can lead back to the first. Two different numbers with this property are amicable, like 220 and 284. The record says Kamal al-Din al-Farisi did not invent this example or Thabit ibn Qurra's older rule; his contribution here is a new factorization approach to proving the rule. What are the divisors of 220?
I can now
Notes
- 1. Ontario Ministry of Education, The Ontario Curriculum: Mathematics
- 2. Weisstein — Amicable Pair, divisors220284
- 3. Weisstein — Amicable Pair, definition
- 4. MacTutor — Kamal al-Din al-Farisi, number-theory
- 5. MacTutor — Kamal al-Din al-Farisi, new-approach
- 6. MacTutor — Kamal al-Din al-Farisi, introduction-and-number-theory
- 7. MacTutor — Kamal al-Din al-Farisi, large-pair-history