Divisor

[huneynumb]

When 391758 and 394915 are divided by a certain three digit number, the three digit remainder is the same in each case. Find the divisor.

[Tartaglia]

divisor 451, remainder 290

[insane_alien]

Tartaglia, please don't just tell him/her the answer. show him/her the method to work it out, even use an example with different digits if you want, but don't just give it to them on a plate.

[YT2095]

WOW! so fast!

 

can I ask How you figured it out?

 

I got the 1 and 5 bit, but was strugling with the rest and didn`t want to plug in all figures and trial/error it as even though I hate maths and no good at it, Cheating is still bad

[Tartaglia]

YT - I factorised the difference 41*11*7 and looked for three figure factors

[YT2095]

huh?

where did you get those figures from?

 

basicly all I did was look at the last digit, and the only thing they had in common was 1, as for the second (10`s place) 5 was the only choice there as nothing goes into 5 other than one and 5 but in the tens position, it had to 5 of the two choices

I got stuck after that

[Tartaglia]

394915 - 391758 = 3157 = 7*11*41 = 7*451

[matt grime]

huh?

where did you get those figures from?

 

Pleaes don't take this the wrong way, but he used elementary/primary/junior school mathematics. Dressing it up we know that we have two numbers X and Y and that divided by some D they have the same remainder R, or as we learnt at elementary/primary/junior school and then forgot when we were given calculators, this means that there are numbers P and Q satisfying:

 

X=DP+R

Y=DQ+R

 

and R is between 0 and D-1

(quotient plus remainder arithemetic, long division, the old fashoined stuff)

 

So, using some little algebraic manipluations it follows that D divides X-Y, or D divides 7*41*11 and is three digits long as is R. This extra information allows you to work out which three digit D and R it is: presumably dividing by 7*41=287 leaves only a 1 or two digit remainder.

[YT2095]

I`m non the wiser now, sorry I asked!

[Tartaglia]

Matt - I am the last of the log generation and seeing people struggling to do simple arithmetic without calculators astonishes me.

My last maths lectures were by one of your colleagues - Judy Holyer in 1983 when I was 18/19

[YT2095]

be astonished then, I can`t use a calculator properly either!

 

I tend to either do it in my head or need pencil/paper, when I "Learned maths" there weren`t any calculators!

 

as I pointed out earlier in this thread, I`m bad at maths, but if you wish to make Fun of that, knock yourselves out

 

it`s probably giving someone else a break.

[Tartaglia]

YT That wasn't a dig at you - merely a retort to matt's comment on calculators

[matt grime]

But YT you do (or did) know how to do this question. It is just long division that you are doing on that bit of paper with a pencil.

 

X/D=P + R/D

Y/D=Q + R/D

 

Is all that the question is saying. Well, where you go from there depends on the person I guess. It's quite a nice question, and if you scribble a few musings it becomes clear what to do - this is somewhere where a lot of students go wrong: they look at the question, throw their hands up and say it's too hard and don't attempt it because they don't 'see' the answer instantly. If they just played around with a few symbols they'd soon realize the questions aren't that hard.

 

OK, you might not have spotted the fact that D must divide the difference between X and Y, but this is one of the equivalent ways of defining modulo arithmetic.

 

Say that X~Y if D divides X-Y. This is equivalent to saying that X and Y have the same remainder on division by D, or that there is an integer E such that X=Y+DE. It's good for people to work that one through the first time they meet it. It's also a shame that teachers don't tend to emphasize that this modulo arithmetic thing is just doing long division again.

 

I have this weird observation about maths:

 

you start off doing long division with remainder until you learn about fractions and then remainders go out the window. Then you learn about the evil decimal expansion and forget about fractions too (I even saw a text book for final year highschool students write sqrt(2)=1.4, come on people!). But then you start to learn that actually you should use 'exact' symbols like pi, e, sqrt and fractions rather than decimals so you start going backwards. By the time you come round to doing maths at university in your first number theory course you're doing, in some sense, long division with remainder again.