Answer:
Explanation:
I’ll start by assuming that we’re working in base 10.
Now then…since 400 isn’t divisible by 9, let’s find the next number up that is.
HINT: If a number is divisible by 9, then its digits can be added together to produce another number that is divisible by nine. The digits in 400 aren’t divisible by 9, since 4 + 0 + 0 = 4. But if you go up to 405, the sum of the digits is divisible by 9. So 405 is the first number from 400 to 40,000 that is divisible by 9.
Using similar logic, we can deduce that 40,000 is also not divisible by 9. What is the highest number below 40,000 that is divisible by 9? If we subtract 1 from 40,000, we get 39,999. The sum of these digits is not divisible by 9 (it’s 39). If we subtract 1 again, we get 39,998. Since the middle three digits are going to be 9s until we get below 39,990, we only have to focus on finding the highest number in which the sum of the first and last digits are divisible by 9. That’s 39,996.
Now we’re really only interested in the numbers from 405 to 39,996 that are multiples of 9, so we can simplify the problem by dividing both numbers by 9, then figuring out the number of included values.
405 / 9 = 45
39,996 / 9 = 4444
And now the question becomes: how many numbers are there from 45 to 4444, inclusive. To find how many numbers there are from a to b, subtract b - a, then add 1 to the result. (You have to add 1, because if you just subtract, you’ll eliminate the lowest number from the list.)
4444 - 45 = 4399
4399 + 1 = 4400
And there’s your answer. There are 4400 numbers between 400 and 40,000 that are divisible by 9.
There’s another way to get this answer: instead of worrying about multiples of 9, let’s figure out how many total numbers there are from 400 to 40,000. You remember how to do that: (40,000 - 400) + 1. There are 39,601 numbers from 400 to 40,000.
Now we can break these numbers up into chunks of 9, where each chunk contains exactly one number that is divisible by 9. That gives us
39,6019=4400.1¯¯¯=440019 chunks.
Since we’re starting with 400, and the first multiple of 9 is 405, then a multiple of 9 will appear in the sixth place in each chunk. But there’s only one-ninth of a chunk left over, which means we won’t get to the multiple of nine in the last piece of a chunk. Therefore, we only run across 4400 multiples of 9 from 400 to 40,000.
What’s that? You want another way to work it out? Okay, no sweat. How many multiples of 9 are less than 40,000? That’s easy enough:
40,0009=4444.4¯¯¯=444449
So there are 4444 multiples of 9 less than 40,000, and we’re 4/9 of the way to the next multiple of 9 when we reach 40,000.
How many multiples of 9 are less than 400? Same procedure:
4009=44.4¯¯¯=4449
There are 44 multiples of 9 less than 400. So if you take all the multiples of 9 that are less than 40,000 (4444) and subtract out the multiples of 9 that are also less than 400 (44), you’re left with 4444 - 44 = 4400 multiples of 9 between 400 and 40,000
I hope that helps!
