Are there more deficient numbers or abundant numbers between 25 and 50?
To determine whether there are more deficient or abundant numbers between 25 and 50, we need to first understand what deficient and abundant numbers are:
- A deficient number is a number for which the sum of its proper divisors (excluding the number itself) is less than the number itself. For example, the proper divisors of 21 are 1, 3, and 7, and their sum is 1 + 3 + 7 = 11. Since 11 is less than 21, 21 is a deficient number.
- An abundant number is a number for which the sum of its proper divisors (excluding the number itself) is greater than the number itself. For example, the proper divisors of 12 are 1, 2, 3, 4, and 6, and their sum is 1 + 2 + 3 + 4 + 6 = 16. Since 16 is greater than 12, 12 is an abundant number.
Now, let's examine the numbers between 25 and 50:
- 25 is a deficient number since the sum of its proper divisors (1 and 5) is less than 25.
- 26 is a deficient number since the sum of its proper divisors (1 and 13) is less than 26.
- 27 is a deficient number since the sum of its proper divisors (1, 3, and 9) is less than 27.
- 28 is a perfect number since the sum of its proper divisors (1, 2, 4, 7, and 14) is equal to 28.
- 29 is a deficient number since it is a prime number (its only proper divisor is 1, which is less than 29).
- 30 is an abundant number since the sum of its proper divisors (1, 2, 3, 5, 6, 10, 15) is greater than 30.
From 25 to 30, we have 3 deficient numbers and 1 abundant number.
- 31 is a deficient number since it is a prime number.
- 32 is a deficient number since the sum of its proper divisors (1, 2, 4, 8, 16) is less than 32.
- 33 is a deficient number since the sum of its proper divisors (1, 3, 11) is less than 33.
- 34 is a deficient number since the sum of its proper divisors (1 and 17) is less than 34.
- 35 is an abundant number since the sum of its proper divisors (1, 5, and 7) is greater than 35.
- 36 is an abundant number since the sum of its proper divisors (1, 2, 3, 4, 6, 9, 12, 18) is greater than 36.
- 37 is a deficient number since it is a prime number.
- 38 is a deficient number since the sum of its proper divisors (1 and 19) is less than 38.
- 39 is a deficient number since the sum of its proper divisors (1, 3, and 13) is less than 39.
- 40 is an abundant number since the sum of its proper divisors (1, 2, 4, 5, 8, 10, 20) is greater than 40.
- 41 is a deficient number since it is a prime number.
- 42 is an abundant number since the sum of its proper divisors (1, 2, 3, 6, 7, 14) is greater than 42.
- 43 is a deficient number since it is a prime number.
- 44 is a deficient number since the sum of its proper divisors (1, 2, 4, 11, 22) is less than 44.
- 45 is a deficient number since the sum of its proper divisors (1, 3, 5, 9, 15) is less than 45.
- 46 is a deficient number since the sum of its proper divisors (1, 2, 23) is less than 46.
- 47 is a deficient number since it is a prime number.
- 48 is an abundant number since the sum of its proper divisors (1, 2, 3, 4, 6, 8, 12, 16, 24) is greater than 48.
- 49 is a deficient number since the sum of its proper divisors (1 and 7) is less than 49.
- 50 is an abundant number since the sum of its proper divisors (1, 2, 5, 10, 25) is greater than 50.
From 31 to 50, we have 12 deficient numbers and 7 abundant numbers.
Based on this analysis, we can conclude that there are more deficient numbers (15 in total) than abundant numbers (8 in total) between 25 and 50.