Hints and solutions to Guess and Check #8

Pythagoras (the right triangle guy) discovered amicable or "friendly" numbers. Two positive integers are amicable if each is the sum of the proper divisors of the other. 284 is amicable with 220. What number is amicable to 1184?

• Let's start out by checking that the given example of 284 and 220 works because the divisors of 284 are 1, 2, 4, 71, 142 and 284. So the sum of the proper divisors of 284 (the ones other than 284) is 1 + 2 + 4 + 71 + 142 = 220. And the divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, 220. So the sum of 220's proper divisors is 1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284.
• For 1184, we first have to compute the sum of its proper divisors. The proper divisors of 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 296, and 592. So the sum of 1184's proper divisors is 1210. We'll leave it to you to find the eleven proper divisors of 1210, and show that they sum to 1184.
• We asked if you could find any more pairs of amicable numbers -- the next few after 1184 and 1210 are
  2620 and 2924
  5020 and 5564
  6232 and 6368
  10744 and 10856

• Many more pairs of amicable numbers have been found (not all numbers have amicable mates). Mathematicians believe that there are infinitely many pairs of amicable numbers, just as there are infinitely many prime numbers, but this has not yet been proved. To learn more, you can do an internet search on the phrase "amicable numbers".

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