Math Is Fun Forum

Use the Law of Sines:

Plug in the different values of b and solve for C.  Now, in order to find a second triangle with that same value for b, you need to see if it's possible to create a triangle with a second C that will preserve the Law of Sines.  To do this simply subtract C from 180.  Determine if this second C can make a triangle or not.

Fundamental Theorm of Arithmetic (in my own words, not from a dictionary):

Every positive integer is either prime or composite.  If it is prime then its only factors are itself and 1.  If it is composite it can be uniquely represented by its prime factors.  For example, 11 is prime, meaning its only factors are 1 and 11.  12 is composite, and can be reduced to 2^2 * 3^1.  No other set of prime factors can represent the number 12.

Formula for calculating phi(n):

Reduce n to it's prime factors -

Now, let's consider these definitions as they relate to your question.  N can be uniquely represented by its prime factors.  Consider the relationship between prime factorization and the formula for phi(n).

Edit: As is generally the case in math, there is more than 1 way to solve this problem.  According to Wikipedia this function has a very simple lower bound:

Given this, it's trivial to prove phi(n) = N for only finitely many n and that phi(n) tends to infinity as n approaches infinity.  Of course, the hard work is actually proving that lower bound (which Wikipedia does not seem to provide), but it's another avenue for you to consider.

78365492 is not divisible by 8.

I'm not much help since it's been some time since I've worked with linear algebra, but thanks to Ricky's hints I think I can help you with #3.

In logic, the contrapositive means this: If A implies B, then Not B implies Not A.  In terms of question 3 it can be reworded like this: If A is not similar to B, then A^2 is not similar to B^2.  This is equivalent to proving that if A^2 is similar to B^2 then A is similar to B.  Note that you don't have to guess the correct answer from the start; either you do prove the assumption (A is not similar to B implies A^2 is not similar to B^2) or you find that it is wrong and prove the reverse by contradiction (my favorite kind of proof, btw).

Now that the idea is laid out, the only work to do is to actually prove the assumption is either right or wrong.  Ricky states outright that the assumption is true, but I don't know of any basic theorems that support that conclusion.  After all, it's not true for the real numbers.  I don't actually know the answer, but I assume you can work from there.

Due to the fact that x+1 must be prime, which I showed above, it's clear that x must be even (or 1, since 1+1 is the only even prime number).  However, I can't see the why y must also be prime.  Could you please show us the justification for this?

You're on the right track, but notice that your grammar requires an equal amount of a's and c's.  What you need is a grammar that produces an equal amount of a's and b's at first, then changes to produce an equal amount of b's and c's.  I'll start you off:

S -> aAb
S -> B
S -> empty
A -> aCbB
A -> empty

I'll leave B and C to you.

To find the remainder we simply need to find the last 3 digits of 9*5^43.  First find the last 3 digits of 5^43, which is fairly simple when you notice the following pattern:

X             Last 3 digits of 5^X
1             005
2             025
3             125
4             625
5             125
6             625
...            ...

For any even number n >= 4, the last 3 digits of 5^n are 625.  For any odd number k >= 3, the last 3 digits of 5^k are 125.  Since 43 is odd we know the last 3 digits of 5^43 are 125.  Multiply this by 9 to get 1125.  This means our remainder when dividing by 1000 will be 125.

What do you mean it's "no PJ"?

I believe ten is the base of our modern number system because we have 10 fingers (barring any deformities).  Imagine living 10,000 years ago with no concept of a number system.  You travel to a nearby village and want to buy some sheep.  How are you going to describe how many sheep you want to buy?  Probably by holding up one finger for each sheep that you want.  The problem comes if you want 12 sheep.  To solve this you flash all ten fingers once, then hold up two fingers.  What you've just done, without knowing you've done it, is count in base ten.  My guess is that from there people were so used to counting in tens that by the time a formal number system was implemented they decided to make it in base 10 for simplicity's sake.

As for what number I would use as a base, I'd go with 16.  Theoretically 1 would be a nice number base because arithmetic operations would be quite simple (e.g., to add 2 numbers together just append them: 000 + 0000 = 0000000), but it would have obvious practicality issues (you couldn't easily record numbers of any reasonable size).  Two could be interesting since it's the smallest prime number and the only even prime number, not to mention it's wide use in modern technology, but it would suffer the same practicality issues that 1 does.

Like using 2 for a number base, 16 would mesh nicely with our use of bits in electronics.  It is a perfect square, and it's root is also a perfect square.  Being a power of 2 it has only a single prime factor.  It would be far more practical from a storage point of view than 1 or 2 (and slightly moreso than ten), but not so big that the number of symbols needed to represent all of the digits would be cumbersome.

It's been a few years since I've done anything with formal languages, but what I think you're looking for is a Type-3 grammar on the Chomsky hierarchy (thank goodness for Wikipedia, I never would have remembered something that detailed).  This means that the language can be generated with a right regular grammar, which means that it can be completely generated with rules of the following type:

A -> b
A -> cD
A -> empty string

For example, let's say that all expressions start with S.  Here's the grammar (or language, I'm unsure of the proper terminology) for the expression a*b*:

S -> aS
S -> bA
S -> empty string
A -> bA
A -> empty string

You can see that by starting with S and following the above rules that we can generate any expression in a*b*.  As another example, let's try to make a list of rules for a(a|b)*:

S -> aA
A -> aA
A -> bA
A -> empty string

Now, with that in mind let's take a look at your examples.  The easiest one would be the second expression: a^n b a^n, n >= 0.  How can we come up with a list of rules for this expression?  Let's start with the beginning a's:

S -> aS

Easy enough.  It's also trivial to add a single b in the middle:

S -> bA

Now we get to the tough part.  How do we make a rule so that we add exactly as many a's as before?  You can see it's plainly impossible, since we have no way of keeping track of how many a's we added at first.  Therefore, this cannot be a regular expression.  This is easily proven for the other expressions as well.  The problem is that they all require you to keep track of how many letter's you've already written, which is not possible for regular expressions.

It's definitely wrong.  Consider this simple example:

According to your method this is equal to

However, it's easy to see that in fact

All that your rearranging accomplished was to show that

which we already knew.