Title
Magic Pairs Problem
Problem Statement The pair of numbers 3 and 37037
is "magic." Here is how the magic works: Pick your favorite digit from among 1,
2, 3, or 9, and multiply it by 3. Multiply the result by 37037. What is the
final answer?
Is the pair of numbers 13 and 8547 "magic?" pick your favorite digit from among
1, 2, 3, or 9, and multiply it by 13. Multiply the result by 8547. What is the
final answer?
How many pairs of "magic" numbers exist?
Problem setup
I multiply 1, 2, 3, or 9 by 3.
Next, that product is multiplied by 33037. What is the answer? What
would allow the numbers 3 and 37037 to be be called "magic"?
I multiply 1, 2, 3, or 9 by 13.
Next, that product is multiplied by 8547. What is the answer? What
would allow the numbers 13 and 8547 to be called "magic"?
Plans to Solve/Investigate the
Problem
I'll take each possible example and
look for a pattern. Then I will extrapolate my results in an attempt to
add more "magic" pairs of numbers.
Investigation/Exploration of the
Problem
1 * 3 * 37037 = 111111
2 * 3 * 37037 = 222222
3 * 3 * 37037 = 333333
9 * 3 * 37037 = 999999
Thus
far, the pattern is clear. Obviously 3 * 37037 = 111111. What would
happen is we tried a few more?
4 * 3 *
37037 = 444444
5 * 3 *
37037 = 555555
13 * 3 *
37037 = 1444443
10 * 3 *
37037 = 1111110
I would
predict that in order to be "magic", the original chosen factor has be to be
less than 10, whether it be even or odd. Beyond the factor 9, the pattern
does not hold true.
Extensions of the Problem
Using a calculator, can you find additional "magic pairs"? How can
your knowledge of patterns help you to find a magic pair?
Author & Contact
rrogers@rockdale.k12.ga.us
