Difference between revisions of "Deficient number"
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| − | A '''Deficient number''' is a number <math>n</math> for which the sum of <math>n</math>'s [[proper divisor|proper factors]] is less than <math>n</math>. For example, 22 is deficient because its [[proper divisor|proper factors]] sum to 14 < 22. | + | A '''Deficient number''' is a number <math>n</math> for which the sum of <math>n</math>'s [[proper divisor|proper factors]] is less than <math>n</math>. For example, 22 is deficient because its [[proper divisor|proper factors]] sum to 14 < 22. The smallest deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, and 17. |
| − | + | ==Problems== | |
| + | ===Introductory=== | ||
| + | ====Problem 1==== | ||
| + | Prove that all [[prime number|prime numbers]] are deficient. | ||
| + | |||
| + | [[Deficient number/Introductory Problem 1|Solution]] | ||
| + | |||
| + | ====Problem 2==== | ||
| + | Prove that all [[perfect power|powers]] of [[prime number|prime numbers]] are deficient. | ||
| + | |||
| + | [[Deficient number/Introductory Problem 2|Solution]] | ||
| + | |||
| + | ==See Also== | ||
| + | [[Perfect number]] | ||
| + | |||
| + | [[Abundant number]] | ||
Latest revision as of 11:39, 9 February 2018
A Deficient number is a number 
for which the sum of 's proper factors is less than . For example, 22 is deficient because its proper factors sum to 14 < 22. The smallest deficient numbers are 1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, and 17.
Problems
Introductory
Problem 1
Prove that all prime numbers are deficient.
Problem 2
Prove that all powers of prime numbers are deficient.
