How would you solve positive integer adding to a certain number questions.

[hirohamada100]

Hey! can anyone help me step by step, because i have absolutely no idea how to do this. i have read the solutions but i did not understand it well enough.
Thanks! Help is appreciated. I found the last bit complicated.

10. What is the last digit of the smallest positive integer whose digits add to 2019?A 1 B 4 C 6 D 8 E 9Solution E

Given two positive integers made up of different numbers of digits, the integer which has thesmaller number of digits will be the smaller of the two integers. [For example, 9 999 999 whichhas seven digits is smaller than 10 000 000 which has eight digits.]Therefore to obtain the smallest postive integer whose digits add up to 2019, we seek positiveintegers with the smallest possible number of digits whose digits add up to 2019. These will bethe positive integers which contains the largest possible number of copies of the digit 9.Now 2019 ÷ 9 = 224 with remainder 3. Therefore the smallest positive integer whose digits addup to 2019 will have 225 digits, of which 224 are 9s, with the other digit being 3.The smallest such positive integer consists of a single 3 followed by 224 copies of 9. That is, it isthe number3224z }| {999 . . . 999 .The last digit of this positive integer is 9.

 

[Dr.Peterson]

Hey! can anyone help me step by step, because i have absolutely no idea how to do this. i have read the solutions but i did not understand it well enough.
Thanks! Help is appreciated. I found the last bit complicated.

10. What is the last digit of the smallest positive integer whose digits add to 2019?A 1 B 4 C 6 D 8 E 9Solution E

Given two positive integers made up of different numbers of digits, the integer which has the smaller number of digits will be the smaller of the two integers. [For example, 9 999 999 which has seven digits is smaller than 10 000 000 which has eight digits.] Therefore to obtain the smallest positive integer whose digits add up to 2019, we seek positive integers with the smallest possible number of digits whose digits add up to 2019. These will be the positive integers which contains the largest possible number of copies of the digit 9.Now 2019 ÷ 9 = 224 with remainder 3. Therefore the smallest positive integer whose digits add up to 2019 will have 225 digits, of which 224 are 9s, with the other digit being 3.The smallest such positive integer consists of a single 3 followed by 224 copies of 9. That is, it is the number3224z }| {999 . . . 999 .The last digit of this positive integer is 9.

I think you're saying that the last paragraph is someone else's explanation, and you don't understand it.

Can you tell us more precisely what part confused you? What is the last thing that you do understand, and what are you unsure about in the next step?

To make that easier, I'll break it into steps:

1. Given two positive integers made up of different numbers of digits, the integer which has the smaller number of digits will be the smaller of the two integers.
2. [For example, 9 999 999 which has seven digits is smaller than 10 000 000 which has eight digits.]
3. Therefore to obtain the smallest positive integer whose digits add up to 2019, we seek positive integers with the smallest possible number of digits whose digits add up to 2019.
4. These will be the positive integers which contains the largest possible number of copies of the digit 9.
5. Now 2019 ÷ 9 = 224 with remainder 3.
6. Therefore the smallest positive integer whose digits add up to 2019 will have 225 digits, of which 224 are 9s, with the other digit being 3.
7. The smallest such positive integer consists of a single 3 followed by 224 copies of 9.
8. That is, it is the number3224z }| {999 . . . 999 .
9. The last digit of this positive integer is 9.

I'm not sure what step 8 is supposed to say. I think something went wrong in your pasting (and you failed to proofread). But apart from that, everything is just what I would have said.

P.S.: A search suggests you got this from here.