"Hi Professor, my algebra teacher multiplied 97 x 103 in her head, and said she "used algebra" to do it. Can you explain what she did? -Anon."

Sure, Anon. There are situations where knowing some algebraic factoring rules can be a big help in doing mental math, and the example your teacher gave is one of those situations.

In order to answer your question, you're going to need to know an algebraic factoring rule called the "difference of squares" rule. If you don't know it, here it is:

a² - b² = (a + b)(a - b)

It's easy to see why this is true, by using the distributive property on the right-hand side:

(a + b)(a - b) = 
a(a - b) + b(a - b) =
a² - ab + ab - b² = 
a² - b²

So to solve 97 x 103 in your head, you need to recognize that 97 is three less than 100, and 103 is three more than 100, which means the following is true:

97 x 103 = 
(100 - 3)(100 + 3) = 
100² - 3² = 
10000 - 9 =
9991

All of that can be done mentally, without writing anything down on paper, and without using a calculator.

Is this a useful technique? Well, it's only useful if you're multiplying two things that are equidistant from something you know the square of.

I happen to know, for example that 16² is 256. So if someone asked me what 15 x 17 is, I would do this:

15 x 17 = 
(16 - 1)(16 + 1) =
16² - 1^{2 =} 
256 - 1 = 
255

And since the only thing I actually did was subtract 1 from 256, I had the answer before you even started reaching for your calculator.

Speaking of 256...since I've done software work ever since I was in high school (back in the 1980s!), and have therefore worked extensively with powers of 2, I have this math fact indelibly imprinted on my brain:

256² = 65,536

I will occasionally use this fact to freak out my math students. I'll write up on the board something like:

246 x 266 = 

And then, while they're still punching that first number into their calculators, I'll finish writing the answer:

246 x 266 = 65,436 

How did I do that in my head? Easy! I deliberately chose two numbers that are equidistant from 256:

246 x 266 = 
(256 - 10)(256 + 10) = 
256² - 10² = 
65,536 - 100 = 
65,436

Of course, the "Difference of Squares" rule is not the only algebraic rule that could help us with mental math. Consider the following question:

Find the prime factorization of 1027.

To solve this mentally, you'll need the following rule for "Sum of Cubes"

a³ + b³ = (a + b)(a² - ab + b²)

I note the following:

1027 = 
10³ + 3³ = 
(10 + 3)(10² - 10 x 3 + 3²) = 
13 x 79

Here's one more example:

Find the prime factorization of 1332.

1332 = 
1331 + 1 = 
11³ + 1³ = 
(11 + 1)(11² - 11 x 1 + 1²) = 
12 x 111 =
(2² x 3) x (3 x 37) = 
2² x 3² x 37

In practical, real world mathematics, these kinds of "stunts" don't usually gain you much, but they can be useful to math teachers in the classroom. If I'm working with students on prime factorizations, I might deliberately create a "sum of cubes" number so that I can find the factorization quickly in my head, and I'll therefore know if my students have the correct answer, without having to go through the work of doing a factor tree. (Yep, I'm lazy that way!)