Question 24

Amicable numbers are a pair of distinct natural numbers (a, b) such that the sum of the proper divisors of a equals b and the sum of the proper divisors of b equals a. Given that (220, y) is a pair of amicable numbers, y equals:

220 = $$2^2\times5\times11$$
So, the Sum of all divisors  = $$(2^0+2^1+2^2)(5^0+5^1)(11^0+11^1)$$
                                         = (7)(6)(12) 
                                         =  504
Proper divisors are all the divisors except number itself .
So, the sum of all proper divisors = $$504-220$$ = 284.

Now to check if the (220,284) are amicable pair of numbers, we have to verify it with checking for sum of proper divisors for 284 : 
284 = $$2^2\left(71\right)$$
So, the Sum of all divisors = $$(2^0+2^1+2^2)(71^0+71^1)$$
                                        = (7)(72)
                                        = 504

Therefore, no of proper divisors of 284 are = 504-284 = 220.

Hence, (220,284) are amicable pair of numbers.

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