I have a problem that goes like this: How many 3 digit numbers have the property that the middle digit is the product of the first and last digits?
I figured that the answer might be 648, but I feel like this is too large of a number. I am probably a couple hundred off. Can someone confirm or help me with this problem?
My work is as follows:
1st digit numbers = 1,2,3,4,5,6,7,8,9
3rd digit numbers = 1,0 since the middle number is less than 10
9*2 = 18 numbers for middle, some repeating
18 * 18 = 324
324 * 2 = 648
How many 3 digit numbers have the property that the middle digit is the product of the first and last digits
combinatoricspermutationsproducts
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Best Answer
Suppose the number begins with $3$. Then it must end with a digit small enough so that the product (which would be the middle digit) is $<10$. That admits $1+[9/3]=4$ choices where $[x]$ is the greatest integer less than or equal to $x$ and the possibility of a zero units digit is included. Do this for all nine possible initial digits and you get
$9+[9/1]+[9/2]+[9/3]+...+[9/9]=32$