Problem: Compute \( 117 \cdot 1001 + 9 \cdot 999 \cdot 13 \).
Let’s try to make this problem into the form \( ab + ac \) so we can conveniently factor out the number \( a \) from both products.
What values of \( a \) should we choose?
It’s not clear at first. For our first product to the right, we can choose to factor out \( 117, 1001 \) or factor each number into prime factors. For our second factor, we can factor out a \( 9 \) or \( 13 \), for instance.
The step of choosing a clever value of \( a \) in common is the most clever part of this problem. For instance, if we have an easier problem \( 2 \cdot 6 + 2 \cdot 7 \), we can factor out the \( 2 \) to change the problem into \( 2 (6+7) \).
Can you figure it out from here?