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Teachers' Lounge Discussion: Integer

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From: Loyd <loydlin@aol.com>
To: Teacher2Teacher Public Discussion
Date: 2002021712:42:20
Subject: Re: integer 2000 digits long.

On 2002021623:18:49, maharage wrote:
>	The left most digit of an integer of length 2000 digits is 3. In
this
>integer, any two consecutive digits must be divisible by 17, or 23.
>There are two possible values for the 2000th digit. What are they?

Answer: It appears to me that the right most digit is 2.  The left
most digit was given as 3.  Two digit multiples of 17 are 17, 34, 51,
68 and 85 and for 23, we have 23, 46,69 and 92.  

So, the number would have to be:

34692 34692 34692 34692 34692 etc. for 2000 digits.  The 2 appears in
the 5th, 10th, 15th, etc. and since 2000 is a multiple of 5, it will
appear in the 2000th place the right most digit.  Using the other
possibility, we have:

34685 17  and that is as far as we can go because there are no
multiples of either 17 or 23 in the seventies.  

Let me know if I am wrong since you said two possibilities.

>Three rugs have a combined area of 200 mē. By overlapping the rugs to
>cover a floor area of 140 mē, the area that is covered by exactly two
>layers of rug is 24mē. What area of floor is covered by three layers
>of rug?

Sorry about this one, I haven't been able to picture a solution.

>
>
>Driving between two towns at 110 km/h instead of 100 km/h saves 9
>minutes. How far apart (in km) are the two towns?  

This is involves a little algebra and the fact that rate x time =
distance (R x T = D):

Let T = time of travel (hours) for the fast car.
    T + 9/60 = time of travel in hours for slow car

Since the distance traveled is the same for both cars, we can write:

R x T = D
110(T)= 100(T+9/60)
110T = 100T + 900/60
10T = 900/60  or 90/6 and
T   = 9/6 hours = 1.5 hours.  

Distance = rate times time, so,
110x1.5 = 165 miles and also, 100(1.5 +9/60) = 165 miles.
  
>
>
>A deck of cards is numbered from 1 to 100. Each card has the same
>number printed on both sides. One side of the card is red, and the
>other side is yellow. All the cards are placed, red side up, on the
>table. First, every card that has a number divisible by two are
turned
>over. Then all the cards that have a number divisible by three are
>turned over. How many cards are red side up?   
>
>

I crossed out all the evens and all divisible by three and the answer
appears to be 33 cards are red side up.  That is:

1,5,7,11,13,17, and so forth.  

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