Abundant Number

Definition of Abundant Number

If the sum of the proper divisors of a number is greater than that number, then that number is called as an Abundant Number.

More About Abundant Number

  • The difference of the sum of the divisors and the number is called ‘abundance’.
  • 945 is the smallest odd abundant number.

Examples of Abundant Number

  • Consider the number 12. The proper divisors of 12 are 1, 2, 3, 4, and 6. Sum of these divisors = 1 + 2 + 3 + 4 + 6 = 16. As the sum of the divisors is 16 and 16 is greater than 12, so 12 is an abundant number.
  • 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100 are the list of abundant numbers between 1 and 100.

Video Examples: Deficient, Abundant and Perfect Numbers 1


Solved Example on Abundant Number

Ques: Which of the following satisfies the abundant number property?

    Choices:
    A. 16
    B. 22
    C. 9
    D. 18
    Correct Answer: D

Solution:

    Step 1: If the sum of the proper divisors of a number is greater than that number, then that number is called as an abundant number.
    Step 2: The proper divisors of 16 are 1, 2, 4, and 8.
    1 + 2 + 4 + 8 = 15, the sum of the divisors of the number 16 is not greater than the number. [15 <>
    Step 3: The proper divisors of 22 are 1, 2, and 11.
    1 + 2 + 11 = 14, the sum of the divisors of the number 22 is not greater than the number. [14 <>
    Step 4: The proper divisors of 9 are 1 and 3.
    1 + 3 = 4, the sum of the divisors of the number 9 is not greater than the number. [4 <>
    Step 5: The proper divisors of 18 are 1, 2, 3, 6, and 9.
    1 + 2 + 3 + 6 + 9 = 21, the sum of the divisors of the number 16 is greater than the number. [21 > 18.]
    Step 6: So, the number 18 satisfies the abundant number property.