Divisors and factors

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

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.

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.

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.

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.

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?

• 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.

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.