Stumped by IQ test: help from uberbrain needed

[rich p]

Marin is thinking of entering Strictly Come Dancing and has promised to sew all his own sequence onto his frock.

 

[vernon]

Is he in prime condition for such a competition?

 

[Stig-OT-Dump]

15 is not prime. It is divisable by 3 and 5.

Whoops - guess who went onlione after getting in from the pub <hangs head in shame>

 

[spire]

They are the so called cyclic numbers.

1 is coprime to itself and the coprime of numbers less than or equal to itself. 2 likewise. 3 is obviously coprime to 2 . 4 has 1 and 3 coprime to itself but 4 and 2 are not coprime so it is junked. 5 is coprime to 1,2,3 and 4 and 4 itself is coprime to 5 so that's all right. If you bother to look at the totient function you'll get all the other numbers including

15 and φ(15)= 8 are coprime.
16 and φ(16)=8 are not. 17 and 6 are. 18 and 6 aren't etc. Quite a few of them stick out like a sore thumb where the totient function is 1 less than n it is obviously coprime or where n is even and the totient function is also even it can be ruled out.

A much better way of doing it is in group theory and cyclic groups. The klein group is isomorphic to the dihedral group of order 4 which is not cyclic so this is not isomophic to the cyclic group of order 4*. So 4 is binned. If you read your group theory which people may or may not be interested in there are dihedral groups of order 2n for 4 onwards so all even numbers bigger than this are binned. If p is a prime number by another theory in abstract algebra then it is a cyclic group. To prove n is not cyclic it is merely enough to find a counter example of a group of order n that is not cyclic. This is time consuming for larger numbers but there are a dozen theorems to help out.

That's what I thought...

...but the published answer is 23????

 

[spire]

Another question in the same test is:

Complete this series:

Bach 4/17, Brahms 6/19, Britten7/20, Beethoven _/_

I believe there are two solutions to this, although only one is published...

 

[ohnovino]

Beethoven 9/18

9 letters in "Beethoven"
Born in the 18th Century


What do I win

 

[yello]

9/22?

 

[adscrim]

I'd go for 9/22 also - number of letter plus 13.

Bach was born in the 18th century.

 

[srw]

They are the so called cyclic numbers.

According to Wolfram:

A cyclic number is an

-digit integer that, when multiplied by 1, 2, 3, ...,

, produces the same digits in a different order. Cyclic numbers are generated by the full reptend primes, i.e., 7, 17, 19, 23, 29, 47, 59, 61, 97, ... (Sloane's A001913).

That certainly rings a bell from the days when I knew anything about number theory, and doesn't correspond to the sequence given. (Point of pedantry - what spire's been given is a sequence, not a series).

1 is coprime to itself and the coprime of numbers less than or equal to itself.

Are you saying that the sequence is the sequence of numbers which are coprime to their totient? If so, how does spire get 23 as the next in the sequence? For what it's worth, if this is a generic IQ test I suspect the question, or spire's typing of it, is a misprint.

[edit]
I've found the wikipedia definition of cyclic numbers, which does go along my's definition. But the next one is 17 - all primes are by definition cylic (using this definition of cyclic). It's a bizarre IQ test which demands knowledge of both composers' birthdates (Bach - at least JS Bach - was born in the 17th century, in 1685) and of group theory.

I think the real answer to spire's question is that it is the list of exercises set for course MA166 at Purdue to be delivered on 28th March:

p733:1,2,3,5,7,11,13,15,23,27

 

[Hover Fly]

I'd go for 9/22 also - number of letter plus 13.

Bach was born in the 18th century.

J.S. was born in 168x, his kids were born in the 18th century.

 

[661-Pete]

I agree entirely. I would go into more Klein theory but I haven't got the bottle.

I'd agree, indeed I'd go further and say, I'm on the same side as you.
In fact it's not possible to be anything else

 

[spire]

[sup]Re the composers:

The the two answers are indeed 9/18 and 9/22, the latter being the published answer.

Re the number sequence:

My typing is correct. (Anyway, I think it unlikely that the solution is as complex as cyclic numbers.)

I am relieved everyone else is baffled. It adds weight to my suspicion that there is a misprint in either the sequence or the answer.
[/sup]

 

[spire]

Another of the questions:

Which is the odd word?

Through, thought, thorough, trough, tough, brought?

 

[ASC1951]

Can anyone answer and explain?

What is the next number in this series?

1, 2, 3, 5, 7, 11, 13, 15,...

The answer is "Whatever number you choose". There is a theorem to this effect - I'll try to dig it out. [I'm not sure whether it works for a sequence of x + n terms, or only for x + 1.]

 

[asterix]

It is the first part of a a series of numbers thus

1+3+5+7+9+11+13+15+17+19

i.e. just add prefix '1' when you repeat the numbers after you have reached 9. AKA adding 10!