Question

A three-digit number is multiplied by a two-digit number whose tens’ digit is 9. The product is a four-digit number whose hundreds digit is 2. How many three-digit numbers satisfy this condition?

Answer 1

There are four 3-digit numbers that meet the conditions: 100, 101, 102, 103

Explanation:

Let's set up the question this way:

We have a 2-, 3-, and 4- digit number to discuss.

• The 2-digit number has a 9 in the tens digit, so it's the numbers 90 through 99

• The 4-digit number has a 2 in the hundreds place.

• The 3-digit number is any 3-digit number (100 through 999)

I want to get a sense as to what we're working with, so let's set the 2 and 3 digit numbers to the smallest and largest values and see what we get:

Smallest

`90xx100=9000`

And we can stop here - we've just eliminated most of the 4-digit numbers. We now know we want to look at 4-digit numbers starting with 92XX.

I think the easiest way to do this is to chart out a multiplication table in the ranges of numbers we're looking at:

`((color(white)(0),100,101,102,103,104),(90,9000,9090,9180,color(brown)9270,9360),(91,9100,9191,color(brown)9282,9373,9464),(92,color(brown)9200,color(brown)9292,9384,9476,9568),(93,9300,9393,9486,9579,9672))`

And this shows us the final result - there are four 3-digit numbers that meet the conditions: 100, 101, 102, 103