Elementary POW, October 28 - November 1,1996
Elem POW Problems || Sept-Nov '96 Problems || Elem POW Main Page
A ten-digit number contains every digit from 0 to 9. The digits are arranged so that the number formed by the first two digits, reading from left to right, is divisible by 2, the number formed by the first 3 digits is divisible by 3, the first 4 by 4, and so on until the whole number is divisible by 10. What is the number? There are two digits that can be placed immediately. Which are they? **Remember, don't just send your answer. Explain the reasoning you used to place the numbers in the proper order.** This week's mentors are from Deborah Bass' Math Methods class at the University of South Carolina Aiken.
The answer to the Problem of the Week (Oct. 28-Nov. 1) was 3,816,547,290 We in the School of Education at the University of South Carolina Aiken were delighted to have correct responses from: Amy Forster of Home School Cygnet, Tasmania, Australia Amy and Andrew Ko Mr. Arnold's fifth grade Northside Elementary, Palmyra, Pennsylvania Eddie Kelly, Ryan Connelly, Kathryn Way, Ryan Lyle, Asheley Cowart, Katie Ussery, Jamie Peper, and Jenny Giles Wilder Middle School Katie Mee of Mrs. Bartosiewicz' 4th grade Titus Elementary, Warrington, Bucks County, PA Krystal Tidwell of McMahon's 6th grade Murray Middle, Ridgecrest, CA Heather Comerci and Andrew Dang Mrs. Geschel's fifth grade Western Salisbury School, Allentown, PA Huw Wilkins, grade 4 Tasmania, Australia Lisa, Zach, Dustin, Carina, Stacey, Claine, and Nicole Mr. Rummel's fifth grade Emerson Elementary (Madison, WI ?) Miss Colwell's sixth grade class re-submitted the correct answer from Oak View Elementary, Fairfax, VA Mrs. Marian Eley's Fourth Grade Class Tri-Village Intermediate School, Hollansburg, OH Michael Scarito of the third grade class Seventh District Elementary Nate Litz, Graham McCorcle, and Justin Weiss Mr. Coulter's third grade math class Forsyth School, St. Louis, MO Sydney Levine Mrs. Bach's 5th grade Mahwah, NJ Udit Garg and Vinay Mr. Nass' 4th grade Georgetown Day School, Washington, DC
The thinking to solve this assignment included the guess and
check strategy; however, many students discovered some other
sound mathematical theories.
Vinay wrote:
I worked on different parts at a time. If the number
was to be divisible by 10, I knew 0 had to be the tenth digit.
The same way 5 had to be in the fifth place. I also figured
out the odd-digits had to be odd and even digits had to be
even [in even/odd position in the number].
The divisibility rule for 8 states that the last 3
digits of the number have to be divisible by 8. So I made a
chart for the digits 6, 7, and 8 and tried all the
combinations leaving out the ones that logically couldn't
work. For example, 8 couldn't be in the eighth place.
For the first three digits I used the
divisibility rule for 3 and also knew that the second digit
had to be even while the other two were odd. I again tried
all the combinations for the digits 1,2,3 and made a chart.
Then I went back and filled in the ninth digit because I knew
it had to be a left over odd number.
After a lot of trial and error I got the answer.
Mrs. Eley's fourth grade class wrote:
We figured out all the odd places would be filled with
odd digits, and all the even places would be filled with even
digits. The only digit that can be in the tenth place is the
0. That leaves the 5 to fill in the fifth place. We listed
all the combinations that would begin with the digit 1 up to
the third place, then all the combinations that would begin
with 7, then with 9. When the combination could not be
divided by three we crossed it out. With the remaining
numbers we added the digit 5 and then listed all the
combinations for the sixth place and tested them by dividing
each by 6. We kept on testing and adding the next place
combinations until we found the one that was correct.
Mr. Rummel's class discovered "along the way that our
calculators could not handle the whole job because they can
only display 8 digits. We switched to a computer calculator
(on a Macintosh) to do the big numbers.
Thanks for your participation!
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