1610.11 – One Million

Tags:Introductory Problems

Find two whole numbers whose product is 1,000,000 but neither of the two numbers has any zeros in it.


Discussion: The whole point of this simply-stated problem is to show the value of making simpler versions of it.

Solving the Problem: How about using 10 instead? That's easy: 10=2×510 = 2 \times 5. OK, how about 100? 100=4×25100 = 4 \times 25, or 22×522^2 \times 5^2.

Oh. So 102=22×5210^2 = 2^2 \times 5^2 . So . . . 1,000,000 is 10610^6, and there we are: 64×15,62564 \times 15,625. Who'd a thunk it?

Heuristic: #14, make a simpler version, big-time.

Using the Problem with Students:

a. as a class activity. We look at the problem together. In good Stella style, it seems hopeless at first. I don't like to spend a lot of time on this one, so I suggest right away that students think of an easier number than 1,000,000. They'll fumble around and I'll keep at them until somebody sees that 10 is a good way to start. Then we quickly go on from there until the solution pops out.

Another way is to do a prime factorization of 1,000,000. It comes down to all 2's and 5's. Lump the 2's together and lump the 5's together and you have it.

Similar Problem: 1180.822: tennis tournament (also in this intro set)