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If the Sum of the Proper Divisors of a Number is Greater Than That Number, Then


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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.




Solved Example on Abundant Number

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 < 16.]

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 < 22.]

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 < 9.]

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.

Related Terms for Abundant Number



  • Addition


  • Divisor


  • Even Number


  • Factors


  • Greater than


  • Odd Number


  • Proper Divisor




Additional Links for Abundant Number



  • Click here for samples


  • Back to Mathematics Dictionary





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