![Page 1: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/1.jpg)
Man Vs. Machine: A Prime Example of Number
Sense
Presented By: Adam Sprague and Stephanie Wisniewski
SUNY Fredonia
AMTNYSOctober 27-29, 2011
![Page 2: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/2.jpg)
Round 1: Repeat or Terminate
𝟏𝟕𝟓
𝟏𝟒𝟎
𝟏𝟐𝟒𝟖𝟓𝟓𝟔𝟗
Terminate
Repeat
Repeat
![Page 3: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/3.jpg)
The Number Sense InvolvedWe recognize a terminating decimal as a fraction with in the denominator. Since the prime factorization of is , the denominator can also be written as where is any integer.
For example:
Since the denominator could be deduced to a power of ten, , we were able to determine this was a terminating decimal.
![Page 4: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/4.jpg)
The Number Sense Involved
A fraction repeats when the denominator cannot be rewritten as a power of ten.
For example:
Due to the uniqueness of prime factorization it will repeat since is not a prime factor of .
![Page 5: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/5.jpg)
Round 2: Is This Number a Perfect
Square𝟖𝟏
𝟖𝟏𝟎
𝟖𝟏𝟎𝟎
𝟖𝟏𝟎𝟎𝟎
No, Not a Perfect Square
Yes, Perfect Square
Yes, Perfect Square
No, Not a Perfect Square
![Page 6: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/6.jpg)
The Number Sense Involved
Every perfect square can be broken down to its prime factors each raised to an even power. We have a perfect square when all the powers of the prime factors are even, such as in , , and .
This is not the case for or because their prime factors are not all of even powers.
![Page 7: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/7.jpg)
The Number Sense Involved
![Page 8: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/8.jpg)
Round 3:Sums of Consecutive Integers
and Perfect Squares
In 2002, the 12th-grade American Mathematics Competition (AMC 12) asked the following problem:
The sum of 18 consecutive positive integers is a perfect square. What is the smallest possible value for this sum?
[http://www.unl.edu/amc/]
![Page 9: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/9.jpg)
Guess & Check Method
18 Consecutive Integers
![Page 10: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/10.jpg)
What If We Asked…
Can the sum of 16 consecutive positive integers be a perfect square?Let us try our guess and check approach;
![Page 11: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/11.jpg)
The Number Sense InvolvedLet represent the consecutive positive integers. The sum of consecutive positive integers is
Now, what can we determine from this product?
![Page 12: Man Vs. Machine: A Prime Example of Number Sense](https://reader030.vdocuments.us/reader030/viewer/2022013101/56813c90550346895da6398a/html5/thumbnails/12.jpg)
The Number Sense Involved
Since and we know that is an odd number, the prime factorization of our sum will always have a factor of .
Since we also know that a perfect square has all prime factors with even exponents we know that it is not possible for consecutive integers to have a sum which is a perfect square.