This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Two distinct positive integers a and b are called amicable numbers (or friendly numbers) if the sum of the proper divisors of a is equal to b, and the sum of the proper divisors of b is equal to a. The proper divisors of a number include all its positive divisors except the number itself. Let (n) denote the sum of all positive divisors of n. Then the sum of the proper divisors of n is (n) - n. So, a and b are amicable if (a) - a = b and (b) - b = a. For example, consider the first amicable pair (220, 284): The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110. Their sum is 1+2+4+5+10+11+20+22+44+55+110 = 284. The proper divisors of 284 are 1, 2, 4, 71, 142. Their sum is 1+2+4+71+142 = 220. Since the sum of the proper divisors of 220 is 284, and the sum of the proper divisors of 284 is 220, the numbers 220 and 284 form an amicable pair. The first 20 amicable numbers (derived from the first 10 amicable pairs, listed in ascending order) are: 220 284 1184 1210 2620 2924 5020 5564 6232 6368 10744 10856 12285 14595 17296 18416 63020 66928 66992 76084 The first 20 friendly or amicable numbers are: 220, 284, 1184, 1210, 2620, 2924, 5020, 5564, 6232, 6368, 10744, 10856, 12285, 14595, 17296, 18416, 63020, 66928, 66992, 76084