Abundant, Deficient, Perfect, and Weird Numbers
Math/Grade 6
Abundant, Deficient, Perfect, and Weird Numbers
Proper factors of a number are all of the factors except for the number itself. For example, the proper
factors of 15 are 1, 3, and 5. Though 15 is a factor, it is not included in the proper factors.
If you add all the proper factors of a number and the sum is less than the number itself, the number is
called deficient. For example, the sum of the proper factors of 16 is 1 + 2 + 4 + 8 = 15 which is less than 16.
If you add all the proper factors of a number and the sum is greater than the number itself, the number
is called abundant. For example, the sum of the proper factors of 20 is 1 + 2 + 4 + 5 + 10 = 22 which is
greater than 20.
If you add all the proper factors of a number and the sum is equal to the number itself, the number is
called perfect. For example, the sum of the proper factors of 6 is 1 + 2 + 3 = 6 which is the same as the
number itself.
Complete the following table:
Number
Proper Factors
Sum of Proper
Factors
Deficient, Abundant or
Perfect?
10
12
18
28
144
200
222
405
496
625
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Abundant, Deficient, Perfect, and Weird Numbers
Math/Grade 6
Revisiting the abundant number 20, it has proper factors of 1, 2, 4, 5, and 10. If you add only the
factors 1 + 4 + 5 + 10, you get 20 (leave out the factor of 2). For most abundant numbers, you can add
some subset of the proper factors to get the original number. However, in some cases, you cannot—
these numbers are called weird numbers. Look at the following abundant numbers and determine if
they are weird numbers.
Number
Proper factors
Subset of factors that add up to
original number?
Weird number?
20
1, 2, 4, 5, 10
1 + 4 + 5 + 10 = 20
No
60
70
80
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Abundant, Deficient, Perfect, and Weird Numbers
Math/Grade 6
Abundant, Deficient, Perfect, and Weird Numbers - Key
Proper factors of a number are all of the factors except for the number itself. For example, the proper
factors of 15 are 1, 3, and 5. The factor 15 is not included in the proper factors.
If you add all the proper factors of a number and the sum is less than the number itself, the number is
called deficient. For example, the sum of the proper factors of 16 is 1 + 2 + 4 + 8 = 15 which is less than
16.
If you add all the proper factors of a number and the sum is greater than the number itself, the number
is called abundant. For example, the sum of the proper factors of 20 is 1 + 2 + 4 + 5 + 10 = 22 which is
greater than 20.
If you add all the proper factors of a number and the sum is equal to the number itself, the number is
called perfect. For example, the sum of the proper factors of 6 is 1 + 2 + 3 = 6 which is the same as the
number itself.
Complete the following table:
Number
Proper Factors
10
12
18
28
144
200
222
405
496
625
1, 2, 5
1, 2, 3, 4, 6
1, 2, 3, 6, 9
1, 2, 4, 7, 14
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100
1, 2, 3, 6, 37, 74, 111
1, 3, 5, 9, 15, 27, 45, 81, 135
1, 2, 4, 8, 16, 31, 62, 124, 248
1, 5, 25, 125
Sum of Proper
Factors
8
16
21
28
259
265
234
321
496
156
Deficient, Abundant or
Perfect?
Deficient
Abundant
Abundant
Perfect
Abundant
Abundant
Abundant
Deficient
Perfect
Deficient
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project
Abundant, Deficient, Perfect, and Weird Numbers
Math/Grade 6
Revisiting the abundant number 20, it has proper factors of 1, 2, 4, 5, and 10. If you add only the
factors 1 + 4 + 5 + 10, you get 20 (leave out the factor of 2). For most abundant numbers, you can add
some subset of the proper factors to get the original number. However, in some cases, you cannot—
these numbers are called weird numbers. Look at the following abundant numbers and determine if
they are weird numbers.
Number
Proper factors
Subset of factors that add up to
original number?
Weird number?
20
1, 2, 4, 5, 10
1 + 4 + 5 + 10 = 20
No
60
1, 2, 3, 4, 5, 6, 10,
12, 15, 20, 30
10 + 20 + 30 = 60
No
70
1, 2, 5, 7, 10, 14, 35
Not possible
Yes
80
1, 2, 4, 5, 8, 10, 16,
20, 40
40 + 20 + 10 + 5 + 4 + 1 =80
No
PUBLIC SCHOOLS OF NORTH CAROLINA State Board of Education | Department of Public Instruction
AIG ~ IRP Academically and/or Intellectually Gifted Instructional Resources Project