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Bmo 2002-2003

http://www.bmoc.maths.org/home/bmomarking.pdf

Question 1:

I found b (that was easy) but couldnt find the others (getting a wrong)

"The neatest way to show that a = 2 is to consider the value of 34!/10^7
modulo 8."

"Last 3 digits (viewed as a number) will be divisible by 8 if the whole number is divisible by 8." Can this be used without proof ? (its used without proof in the book)

for c and d:

34! is divisible by 9 and 11, so sum of digits is divisble by 9, so c + d - 3 is divisible by 9, and for 11|34!, c - d - 8 is divisible by 11.

I cant see where " c + d - 3" and "c - d - 8" come from. We dont add up all the other digits and find out what this is modulo 9 or 11 do we?

Or do we? I'm not sure :rolleyes:
Reply 1
Avatar for DFranklin
DFranklin
18
Do you know a "digit test" to tell if a number is divisible by 9?
Do you know a "digit test" to tell if a number is divisible by 11?

(This is for the c,d bit).
Reply 2
Avatar for refref
refref
OP
Yes. We then add up all the other digits dont we? Hmmm. Well I suppose there is plenty of time :smile:

Are we allowed to use these divisiblity tests without proof?
Reply 3
Avatar for DFranklin
DFranklin
18
As for finding a: the hint they give is a little cryptic to my mind. Can you show that 8 divides 34! / 10^7? (And can you see how to use this to find a)?
Reply 4
Avatar for DFranklin
DFranklin
18
refref
Yes. We then add up all the other digits dont we? Hmmm. Well I suppose there is plenty of time :smile:
That works for 9. But I think we can agree that 11 is divisible by 11, and adding up the digits gives 2...

(You may want to google for divisibility tests for 11).

Are we allowed to use these divisiblity tests without proof?
I would think so - I dare say SimonM or others will pipe up if I'm wrong.
Reply 5
Avatar for refref
refref
OP
DFranklin
That works for 9. But I think we can agree that 11 is divisible by 11, and adding up the digits gives 2...

(You may want to google for divisibility tests for 11).

I would think so - I dare say SimonM or others will pipe up if I'm wrong.



Hmm? I don't really understand. I thought you would take 2 - 9 + 5........until we get to a (which we would have already found out?).

Ahhh I'm confused now.
Reply 6
Avatar for DFranklin
DFranklin
18
refref
Hmm? I don't really understand. I thought you would take 2 - 9 + 5........until we get to a (which we would have already found out?).

Ahhh I'm confused now.
No, that's fine. But what you said was "we add up all the other digits", which isn't a valid way of testing for divisibility by 11.
Reply 7
Avatar for refref
refref
OP
DFranklin
No, that's fine. But what you said was "we add up all the other digits", which isn't a valid way of testing for divisibility by 11.


Oh whoops. Okay thanks.

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