We are told to assign a number to each letter of the alphabet corresponding to the letter's position in the alphabet (a=1, b=2, c=3, etc.). Using this cipher, we must find a familiar first name, the letters of which have corresponding numbers whose product is 100,000. Now, we could simply guess names until we got an answer, but then this wouldn't be much of a brain teaser. Instead, we are going to eliminate all the letters of the alphabet that we can't use, narrowing down our options.

In order for a letter to be usable, it needs to be possible for it's corresponding number to be part of a product equal to 100,000. If this is possible, that number is called a factor of 100,000. Determining all possible factors of 100,000 is actually going to be fairly easy if we use a handy tool called the fundamental theorem of arithmetic. This theorem says that any whole number greater than one can be broken down into a unique product of prime numbers. So, we want to look for prime numbers that are factors of 100,000.

First, notice that 100,000 = 10 x 10 x 10 x 10 x 10.

Since 10 = 2 x 5, and since 2 and 5 are both prime numbers, we can break down all five of those 10's into 2's and 5's, which, after rearranging the numbers, gives us

100,000 = 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5

This not only tells us that 2 and 5 are both factors, but so is any number which is equal to the product of some combination of those 2's and 5's. Since the only factors we are interested in are those between 1 and 26 (anything higher won't have a letter assigned to it), this leads to the following list of letters:

a = 1

b = 2

d = 4 = 2 x 2

e = 5

h = 8 = 2 x 2 x 2

j = 10 = 2 x 5

p = 16 = 2 x 2 x 2 x 2

t = 20 = 2 x 2 x 5

y = 25 = 5 x 5

Notice that I included the letter a in that list, even though it isn't a product of 2's or 5's. This is because no matter what factors we use in our product of 100,000, multiplying that product by 1 will still leave us with 100,000. So, we can use the letter a as much as we want without it messing anything up.

So, we've managed to significantly reduce the number of letters we can use for our name. This will make finding the right name much easier. However, we have to be careful. This is merely a list of candidate letters, we can't just come up with any name using them. We still need to check that their product is equal to 100,000. With that said, a familiar first name which works for this problem is Betty:

2 x 5 x 20 x 20 x 25 = 100,000