What is is 21 and 18 weird?

Is 21 and 18 Weird Numbers?

Let's examine whether the numbers 21 and 18 are considered weird numbers. Remember, a weird number is a number that is abundant (the sum of its proper divisors is greater than the number itself) but is not a semiperfect number (it cannot be expressed as the sum of some or all of its proper divisors).

Analyzing 21:

• The proper divisors of 21 are 1, 3, and 7.
• The sum of these divisors is 1 + 3 + 7 = 11.
• Since 11 < 21, 21 is not abundant.
• Therefore, 21 cannot be a weird number.

Analyzing 18:

• The proper divisors of 18 are 1, 2, 3, 6, and 9.
• The sum of these divisors is 1 + 2 + 3 + 6 + 9 = 21.
• Since 21 > 18, 18 is abundant.
• Now, we need to check if 18 is semiperfect. Can we find a subset of its proper divisors that sum to 18?
  • 1 + 2 + 3 + 6 + 9 = 21
  • 9 + 6 + 3 = 18. YES.

Since 18 is abundant and semiperfect, it is not a weird number.

Conclusion:

Neither 21 nor 18 are weird numbers. 21 is not abundant, and 18, while abundant, is semiperfect.