Man Vs. Machine: A Prime Example of Number

Sense

Presented By: Adam Sprague and Stephanie Wisniewski

SUNY Fredonia

AMTNYSOctober 27-29, 2011

Round 1: Repeat or Terminate

𝟏𝟕𝟓

𝟏𝟒𝟎

𝟏𝟐𝟒𝟖𝟓𝟓𝟔𝟗

Terminate

Repeat

Repeat

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.

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 .

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

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.

The Number Sense Involved

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/]

Guess & Check Method

18 Consecutive Integers

What If We Asked…

Can the sum of 16 consecutive positive integers be a perfect square?Let us try our guess and check approach;

The Number Sense InvolvedLet represent the consecutive positive integers. The sum of consecutive positive integers is

Now, what can we determine from this product?

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.

In Conclusion…Thank you.

Adam Sprague Stephanie Wisniewskiasprague@fredonia.edu swisniewski@fredonia.edu

SUNY Fredonia