Question

Show that 1184 and 1210 are amicable numbers.

Answer 1

Hint: To solve this question, firstly we will see what are actually amicable numbers by definition and using some examples. After that, we will factorize the numbers 1184 and 1210 and then, we will add the factors of 1184 and factors of 1210 individually. And, then using the definition of amicable numbers we will see whether 1184 and 1210 are amicable numbers or not.

Complete step-by-step solution:
Now, let us see what amicable numbers are.
Amicable numbers are two different numbers related in such a way that the sum of the proper divisors of each is equal to the other number.
Now, let's see some examples. The smallest pair of amicable numbers which is known is (220, 284). They are amicable because the proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, of which the sum is 284. And similarly, the proper divisors of 284 are 1, 2, 4, 71, and 142 of which the sum is 220.
Now, proper divisors of number 1184 are 1, 2, 4, 8, 16, 32, 37, 74, 148, 269, 592 and similarly proper divisors of 1210 are 1, 2, 5, 10, 11, 22, 55, 110, 121, 242, 605.
Now, if we add proper divisors of number 1184, we get sum equals to 1210, that is
 1 + 2 + 4 + 8 + 16 + 32 + 37 + 74 + 148 + 269 + 592 = 1210
and, if we add proper divisors of number 1120, we get sum equals to 1184, that is
 1 + 2 + 5 + 10 + 11 + 22 + 55 + 110 + 121 + 242 + 605 = 1210
Hence, numbers 1184 and 1210 are amicable numbers.

Note: A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3. Remember the definition of Amicable numbers and always remember that Amicable numbers are in pairs. Also, do not leave any factor of number as this may change the result for the pair of numbers.