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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

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 < 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 <>
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.

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