# Secret of the Universe

**Secret of the Universe #1**

**Hidden within the prime factorization
of any whole number is
its complete set of factors.**

**Hint: Look for them in pairs.**

**Secret of the Universe #2**

**The number of factors
in the complete set can be found by:
**

**Secret of the Universe #3**

**The LCM and GCF of two numbers
x and y
are related by the equation :**

**Secret of the Universe #4
Patterns in Factorizations**

**The prime factors of perfect squares
Will always have:**

**Similarly, The prime factors of
perfect cubes Will always**

**The prime factors of powers of ten
Will always be**

Remember that a factorial has been defined

as follows:

6! = 6x5x4x3x2x1

10! = 10x9x8x7x6x5x4x3x2x1

30! = 30x29x28x27x…x3x2x1

60! = 60x59x58x57x…x3x2x1

If each of the above factorials was written out as

a standard numeral, how many of the last place

values would be occupied by consecutive

zeroes?

Mrs. DeYoung has chartered a chapter of BUNGEE

(Big Unwieldy Number Groups with Excessive Exponents).

The membership requirements for any number have been

established as follows:

1. Only even numbers may be admitted to membership.

2. Each member must be a multiple of 12.

3. Each member must be a perfect square .

4. Each member must possess least 1000 factors .

5. If a member chooses to appear as a standard numeral, the

member is responsible for enforcing the proper “dress

uniform.” (“Dress uniform” is decided by the number of zeroes

at the end of the line . Exactly four terminal zeroes must be

showing at all times to be proper .)

Your task, as representatives of the membership committee,
is

to determine whether the following number groups satisfy the

BUNGEE membership requirements. For applicants that fail to

meet the requirements on the first try, please give them

appropriate constructive criticism. Explain to them exactly

which primes they should “recruit” prior to their reapplication.

Please note that Hope College is committed to nondiscrimination

in applying membership status to any number

groups, so your constructive criticism should never suggest

that any prime factors already present be eliminated .

Applicant A: 2^{8}x3^{4}x5^{4}x17^{4}

Applicant B: 2^{2}x7^{8}x11^{4}x17^{6}x23^{5}

Applicant C: 2^{4}x7^{8}x13^{4}x19^{2}

Applicant D: 2^{4}x3^{8}x5^{4}x13^{2}x97

Applicant E: 3^{5}x5^{7}x11^{4}x13^{2}x23

Applicant F: 2^{6}x3^{2}x5^{4}x7^{4}x23^{6}

**Prime Factor Bingo**

A smaller number |
A number with different parity (odd/even) |
A number with exactly 3 matching prime factors |

A number with exactly 1 matching prime factor |
FREE | A larger number |

A number with NO matching prime factors |
The same number |
A number with exactly 2 matching prime factors |

Materials needed: Students each receive the same set of
cubes from which to

select exactly five to make a “special number.” These five are then stacked

(multiplied) to make the individual student’s numbers. I usually give each
student

a stack of 5 red, 5 blue, 2 green and 2 yellow cubes (or 2^{5}x3^{5}x5^{2}x7^{2})

Some possible choices:

2x2x2x2x5 = 80

2x3x3x5x7 = 630

2x3x5x7x7 = 1470

The play: Students circulate to compare numbers and
initial each other’s papers

in the appropriate spaces. Each space must contain a different signature . The

word “matching” means a one-to-one match of cubes (or primes).

Some discussion questions:

What is the largest possible “5-cube number”?

What is the smallest possible “5-cube number”?

How are the primes related to what makes individual numbers larger or smaller?

Do more people have odd numbers or even numbers? Why?

Could any of our numbers be perfect squares ?

Could any of our numbers be powers of ten?

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