Students will learn why the following theorems are true:

$$\begin{align} & \text{gcd}(xy, xz) = x\text{gcd}(y, z) \\ & \text{lcm}(xy, xz) = x\text{lcm}(y, z) \end{align}$$If we're interested in finding the GCD or LCM of \(x\) and \(y,\) but finding the prime factorization of \(x\) and \(y\) would be tedious, these theorems will be useful. For example, we can use the theorems to find the GCD or LCM of 96968 and 72726.

Students will learn why the following theorems are true:

$$xy = \text{gcd}(x, y)\text{lcm}(x, y)$$NOTE: This is not covered by the Common Core, nor Khan Academy.

Next, give your students these challenges:

- Cubes Cut Into Four Pieces by NRICH

Make the equation below true by replacing each letter with a unique digit (0-9).

$$P + P + P = I = G + G$$Here's the solution:

\(P + P + P = I,\) and \(I = G + G.\) Thus, \(I\) must be divisible by 3 and 2. The only single digit number with this property is 6. Now we have \(P + P + P = 6 = G + G,\) so \(P\) is one-third of \(I,\) and \(G\) is one-half. So \(P = 2\) and \(G = 3.\) In conclusion, the equation is \(2 + 2 + 2 = 6 = 3 + 3.\)

Conclude by leading this investigation:

Packing Pasta (estimate, measure, volume)

by MathPickle