A Multiplication Based Logic Puzzle

Posts tagged ‘Pattern’

378 Do You See a Pattern?

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?

2nd Multiplication Pattern

The pattern looks like this on the multiplication table:

Basic Multiplication Table Pattern

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

 

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361 Do You See a Pattern?

When 2^361 is divided by 361, the remainder is 116, not 2. That means that 361 is definitely a composite number. Its factors are listed at the end of this post.

361 isn’t used as often, but it is just as special as some of the numbers in the table below:

Multiplication Pattern

The pattern can also be seen along the diagonals in this ordinary multiplication table:

Multiplication Table Pattern

This pattern could be very helpful to students who are learning to multiply. I have seen plenty of students who knew 7 x 7 = 49, but couldn’t remember what 6 x 8 is.

Years after I learned the multiplication facts, I learned how to multiply binomials in an algebra class. I learned about the difference of two squares. In the example below one of the squares is n² and the other square is 1² which is equal to 1. I learned that the equation

(n-1)(n+1)

is true for ALL numbers, but nobody pointed out any practical examples to make it more meaningful. The table at the top of the page contains twelve practical examples. Let’s see how you do applying it to products of a few larger numbers.

Sometimes we find easy ways to remember certain products like
13 and 14 squared

We can use those products to help us remember other products easily by applying the difference of two squares. Try these: (Yes, you can easily do them without a calculator!)

  • 13 x 13 = 169. How much is 12 x 14?
  • 14 x 14 = 196. How much is 13 x 15?
  • 20 x 20 = 400. How much is 19 x 21?
  • If you know that the first ten powers of 2 are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, then it’s easy to remember that 16 x 16 = 256. How much is 15 x 17?
  • What if we go in the opposite direction…It isn’t too hard to multiply 18 x 20 in your head to get 360. How much is 19 x 19?
  • 22 x 20 = 440 was also easy to find. How much is 21 x 21?
  • 30 x 30 = 900. How much is 29 x 31?
  • 100 x 100 = 10,000. How much is 99 x 101?

Multiplication Pattern 2

Did you figure out what 361 has to do with this pattern? It is a perfect square just like 1, 4, 9, 16, and 25. Here is its factoring information:

  • 361 is a composite number.
  • Prime factorization: 361 = 19^2
  • The exponent in the prime factorization is 2. Adding one we get (2 + 1) = 3. Therefore 361 has exactly 3 factors.
  • Factors of 361: 1, 19, 361
  • Factor pairs: 361 = 1 x 361 or 19 x 19
  • 361 is a perfect square. √361 = 19

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