### Brain Teasers

# Weird Numbers

12 is an abundant number, because the sum of its proper divisors: 1+2+3+4+6=16, and this is larger than the number itself. However, you will notice that 1+2+3+6=12.

A weird number is an abundant number, n, for which no subset of its proper divisors adds to n. What is the smallest weird number?

A weird number is an abundant number, n, for which no subset of its proper divisors adds to n. What is the smallest weird number?

### Hint

They are very rare indeed. The smallest is somewhere between 50 and 100.### Answer

The smallest weird number is 70.The next three being 836, 4030 and 5830.

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## Comments

A toughie, especially without the hint! One thing that helps is noting that no number that is prime or a product of two primes can be abundant, and that if n is an abundant, non-wierd number, then so is 2*n (because you can form it as n + the factors that added up to n).

That focuses the search a little - if the number is even, it must be 2 times a non-abundant number, and if it's odd, it needs to be pretty big to be abundant.

That focuses the search a little - if the number is even, it must be 2 times a non-abundant number, and if it's odd, it needs to be pretty big to be abundant.

Sometimes I wish I wasn't so dumb and that I would be pleased by understanding and finding pleasure in such complicated maths problems ..

You guys are geniuses !

You guys are geniuses !

I see what you mean, I can't believe after 10 votes half of the people voted this easy, or else no vote.

Two points:

1. You defined a weird number as an abundant number, and you defined an abundant number as one whose proper divisors add to a number greater than the number itself. The proper divisors of 70 are: 1, 2, 5, 7, and 35 (except, see point # 2), which add to 50; thus, 70 is not an abundant number.

2. You're forgetting that the number itself is a proper divisor. Since you included 1 as a divisor, you should also include 70 (0r 12, in your example). Thus, there are no weird numbers, since one of the subsets of the divisor is the number itself.

1. You defined a weird number as an abundant number, and you defined an abundant number as one whose proper divisors add to a number greater than the number itself. The proper divisors of 70 are: 1, 2, 5, 7, and 35 (except, see point # 2), which add to 50; thus, 70 is not an abundant number.

2. You're forgetting that the number itself is a proper divisor. Since you included 1 as a divisor, you should also include 70 (0r 12, in your example). Thus, there are no weird numbers, since one of the subsets of the divisor is the number itself.

Sorry, forgot the 10, but that still only adds up to 60.

Unless the number is square, it will always have an even number of divisors. It is always useful to list divisors in pairs that multiply to make the number. So for 70: 1x70 2x35 5x14 and 7x10. That way, you don't miss any divisors. You will then notice that the proper divisors are 1,2,5,7,10,14 and 35, which add to 74, making 70 abundant. As no subset of these numbers add to 70, and 70 is the first abundant number for which this is true, it IS the first weird number.

Enjoyed it, got it with some work. Thanks!

theres really numbers that are called"weird" to me thats wierd! LOL

Good teaser although I had no idea about it

Good teaser although I had no idea about it

But because these are known facts, they can be easily found with a little research. I think this is more like trivia than a mathematics teaser. Sorry.

From Wikipedia...

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ..

The first few weird numbers are 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, ..

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