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If the Sum of the Proper Divisors of a Number is Greater Than That Number, Then That Number is Called as an Abundant Number

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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

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.

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

About the Author

Why do people's name and number come through as unknown on the caller id?

I have the anonymous call rejection turned on and yet some numbers manage to pass through as unknown name and unknown number. Is there any way to fix this?

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