A Fraction and a Prime Number Question

Date: 9 Feb 1995 15:07:05 -0500
From: Pat Couch
Subject: Kevin's Question

Dear Dr. Math,

I have two questions I would like some help with. 
Here is the first question:

If 1/4 of a number equals 1/20, then 5/4 of this same 
number equals
a) 1/4
b  1/5
c) 1/16
d) 1/64

I'm not sure how to figure out the answer. 

2. The product of the first 100 prime numbers is 
not divisible by:
a) 44
b) 55
c) 66
d) 77

Could you please help me with this one too.

Thanks. Kevin Richardson

Date: 12 Feb 1995 02:13:53 -0500
From: Dr. Ken
Subject: Re: Kevin's Question

Hello there!

Well, here's what we can do.  Let's translate your first 
sentence into an algebraic statement (that just means we'll 
let letters stand for numbers, basically).  If we let x
represent the number in question, we get 

  1            1
 --- x X  =  ----
  4           20

Then we think about what we're trying to find out.  We're 
trying to find a value for the quantity (5/4) times X, right?  
So what happens when we multiply the equation by 5?  
We get

  5            5         5         1 
 --- x X  =  ----  =  -------  =  ---
  4           20       4 x 5       4

So that's our answer, 1/4.  Did this make sense?  If you 
still don't understand, write back to us.

This problem is a little more interesting.  Let's look at a 
list of the first few prime numbers:

2, 3, 5, 7, 11, 13, 17, 23, 29, ...

Now, it looks at first like we're going to have to find out 
what the product of the first 100 prime numbers is, and 
then divide it by the four options we're given.  But this 
seems too hard, so there must be some kind of trick.
And in fact, there is.

What do you notice about the list of primes there?  How 
many of them are even numbers (divisible by 2)?  Will 
any prime number except 2 be even?  The answer is no, 
and I'll leave it up to you to figure out why (hint: what's
the definition of a prime number?).

Now look at the first option we're given:  44.  If a 
number is divisible by 44, that means it's divisible by 4 
and 11, since 4 x 11 = 44.  But if we mulitiply any 
number of prime numbers together, the only one that's 
going to give us any twos is 2, because it's the only even 
number among the primes.  So we couldn't possibly get 
_two_ twos (i.e. 4) by only multiplying together
prime numbers.

I hope this helps you understand these problems.  
They're neat once you understand them!

-Ken "Dr." Math