Most people know that 378 is a composite number because it’s obviously divisible by 2, but I know that 378 is a composite number because 18 x 21 = 378. I didn’t use a calculator or multiply 18 and 21 by hand. I know this multiplication fact because I can see a pattern, because 18 + 3 = 21 and because SUBTRACTING 2 from a number is easy.
“What is this pattern?” you ask. Here it is. Do you see the pattern, too?
The pattern looks like this on the multiplication table:
So how did I know that 18 x 21 = 378? I knew because I can easily compute 19 x 20 = 380 in my head, and 380 – 2 = 378.
What is all the factoring information for 378?
- 378 is a composite number.
- Prime factorization: 378 = 2 x 3 x 3 x 3 x 7, which can be written 378 = 2 x (3^3) x 7
- The exponents in the prime factorization are 1, 3, and 1. Adding one to each and multiplying we get (1 + 1)(3 + 1)(1 + 1) = 2 x 4 x 2 = 16. Therefore 378 has exactly 16 factors.
- Factors of 378: 1, 2, 3, 6, 7, 9, 14, 18, 21, 27, 42, 54, 63, 126, 189, 378
- Factor pairs: 378 = 1 x 378, 2 x 189, 3 x 126, 6 x 63, 7 x 54, 9 x 42, 14 x 27, or 18 x 21
- Taking the factor pair with the largest square number factor, we get √378 = (√9)(√42) = 3√42 ≈ 18.442
This must be the kind of shortcut that “math geniuses” use to impress people by doing advanced mental arithmetic. It’s a bit like being a magician.
Well, I suppose if you’d twisted my arm I’d have said I was aware of this, but I’ve never ever thought of capitalising upon it, even though if I were faced with 19×21 I’d immediately say it must be 400-1 = 399; and that 18×22 will be 400-4=396.
I’ve met a 9 year old who recognized this pattern on his own as well as other patterns in the multiplication table. His peers consider him to be a math genius for that and other reasons. Magicians know tricks not everyone else knows. I’m sure there are many other people who’ve known this pattern from their youth and even occasionally capitalize on it with or without arm twisting!
Recognizing patterns is a very important skill for mental calculations and mathematics in general. This is especially more apparent in higher mathematics.
As for mental calculations, there are so many techniques available for a mental calculator but we should be able to quickly recognize when those techniques are applicable and what method is the most efficient for solving a particular problem.
Very well said!