Anonymous

Anonymous asked:

How many positive integers less than 1,000 do not have 7 as any digit? and How many positive integers less than 1,000 are divisible by 7? thank you!

Ah. Two counting questions!

If it’s cool with you, I’m going to answer them in reverse order, since the first one is a bit more involved. 

To find the number of positive integers less than 1,000 that are divisible by 7, simply figure out the greatest multiple of 7 that’s less than 1,000. 

1000/7 = 142.857…, so the greatest multiple of 7 that’s less than 1,000 is going to be 7 × 142 = 994. More importantly, that means there are 142 multiples of 7 less than 1,000.

If, as I suspect, you meant to ask how many positive integers less than 1,000 are NOT divisible by 7, simply take the number of positive integers less than 1,000 (which is 999) and subtract the 142 you know to be multiples of 7. (Shaded region questions aren’t the only questions where this technique is useful!)

999 - 142 = 857.

Onto your second question. To find the number of positive integers less than 1,000 that don’t have 7 as any digit, we’re going to break the integers into 3 categories: 1-digit, 2-digit, and 3-digit integers. (There are other ways to do this, of course, but I want to walk you through it this way to point out a common error. You’ll see.) 

First, how many 1-digit positive integers don’t have 7 as a digit? Well, we’ve got 1, 2, 3, 4, 5, 6, 8, and 9. In total, 8 integers. Note: 0 isn’t a positive integer, so we don’t count it. 

Now, how many 2-digit integers don’t have 7 as a digit? If you’re following along, draw two hangman blanks, like so:

____ × ____

We’re going to fill those in with the number of possibilities for each place. The tens place has 8 possibilities (we still don’t count 0, because if the tens place was 0, we’d really have a one-digit number. The ones digit could be anything but 7, including 0 (60 would count, for example), so there are 9 possibilities there.

8 × 9 = 72

Finally, we’ll deal with the 3-digit integers. You probably see how this is going to go by now.

____ × ____ × ____

There are 8 possibilities for the hundreds place (again, to make sure we’re really dealing with a 3-digit integer). 9 possibilities each for the tens and ones places. 

8 × 9 × 9 = 648

Now we just need to add those up, and we’ll have our answer.

8 + 72 + 648 = 728

Cool, right? Of course, there is a shortcut, but it’s got a major pitfall. We might have started with 3 blanks right away, instead of breaking things out like above.

____ × ____ × ____

Now, since we’re doing it all at once, the hundreds place (and the tens place, and the ones place) are allowed to accept 0s. 

9 × 9 × 9 = 729. Wait! WHAT? How come we’re getting a different answer?

Well, when we allow each place to be zero, we count the possibility that all three will be 0. 000 is not a positive integer, so we don’t want to count it, but our shortcut doesn’t know that. So we need to remember it, and subtract 1 at the end to get the correct answer. 

9 × 9 × 9 – 1 = 728

There. That’s better.

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