Grades 11-12 Video Solutions 2021-video 2021 11-12/16

Video Transcription

Problem number 16 states, for a positive integer n, we denote it by p of n, the product of the digits of n, written in decimal form.  For example, p of 23 is equal to 2 times 3, which is equal to 6. What is the value of the sum p of 10 plus p of 11 plus p of 12, on and on and on, through p of 99 plus p of 100?  So if we have an integer with digits a, b, c, the p function will evaluate to a times b times c.  So the first number is 10, so if we get 10, then the p function will return 1 times 0, which is 0, since anything times 0 is 0.  It will do this for all numbers that have a 0, so we will disregard them, as they will contribute nothing to our overall sum. So first let's start off with 11.  11 will give us 1 times 1, which is 1. 12 will give us 1 times 2, which is 2. 13 will give us 1 times 3, which is 3, all the way up through 19.  So for the sum of the p values of the numbers between 11 and 19, we get 1 plus 2 plus 3, on and on and on, plus 8 plus 9. So the sum we end up getting is 45.  Keep this sum in mind, because it comes up a lot in this problem. Now let's try between 21 and 29. So we get 2 times 1, plus 2 times 2, plus 2 times 3, and so on through 2 times 9.  You may notice we can factor out a 2, and we get the previous sum, which was evaluated to 45, multiplied by 2.  Between 31 and 39, we would get the same thing, just multiplied by 3 this time. So now that we found the pattern, let's calculate the sum.  So the sum of the p values of the numbers between 10 and 19 is going to be 45. Sum of the p values between 20 and 29 is 2 times 45. Between 30 and 39 is 3 times 45.  And so on through 90 through 99, which is going to be 9 times 45. So let's add up all of these sums to get one final sum. And let's factor out a 45.  And we get our favorite sum again. So we get 45 times 45, which is going to be equal to 2,025, our final answer.  So the question asked us, what is the value of the sum of p of 10 plus p of 11 plus p of 12, on and on through p of 19 plus p of 100? The answer is 2,025. Answer A.

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