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How many nonzero digits are there in the integers from 1 to 1000?
eg 101 = 2 non zero digits
102 = 2 nonzero digits
509 another 2 nonzero digits
500 1 nonzero digit
how many in total?
and the answer is nowhere near 900
3 Answers
- Dragon.JadeLv 78 years ago
Hello,
= = = = = = = = = = = = = = = = = = = =
Consider all one digit numbers:
= = = = = = = = = = = = = = = = = = = =
From 0 to 9 (we might as well include the initial zero since it will logically be eliminated in the result, this will be useful later...)
1, 2, 3, ... and 9
So we obtain a total of 9 non-zero digits.
= = = = = = = = = = = = = = = = = = = =
Consider all two digits numbers:
= = = = = = = = = = = = = = = = = = = =
From 10 to 19, you obviously have 10 times the digit "1" and "all the one digit numbers" above!
So we obtain a total of 10+9=19 non-zero digits.
From 20 to 29, you obviously have 10 times the digit "2" and "all the one digit numbers" above!
So we obtain a total of 10+9=19 non-zero digits.
....
From 90 to 99, you obviously have 10 times the digit "9" and "all the one digit numbers" above!
So we obtain a total of 10+9=19 non-zero digits.
So the total for all two-digits integers is:
19 + 19 + 19 + ... + 19 = 9 × 19 = 171
Adding in the 9 from the one-digit integers:
Number of non-zero digits from 1 to 99 is 171+9=180
= = = = = = = = = = = = = = = = = = = =
Consider all three digits numbers:
= = = = = = = = = = = = = = = = = = = =
From 100 to 199, you obviously have 100 times the digit "1" and "Number of non-zero digits from 1 to 99" above!
So we obtain a total of 100+180=280 non-zero digits.
From 200 to 299, you obviously have 100 times the digit "2" and "Number of non-zero digits from 1 to 99" above!
So we obtain a total of 100+180=280 non-zero digits.
....
From 900 to 999, you obviously have 100 times the digit "9" and "Number of non-zero digits from 1 to 99" above!
So we obtain a total of 100+180=280 non-zero digits.
And logically, you get:
Number of non-zero digits from 100 to 999 is 9×280=2520
Adding in the "Number of non-zero digits from 1 to 99" above:
Number of non-zero digits from 1 to 999 is 2520+280=2800
= = = = = = = = = = = = = = = = = = = =
Then we need to add one "1" digit from 1000 and finally obtain:
= = = = = = = = = = = = = = = = = = = =
Conclusively, the number of non-zero digits from 1 to 1000 is:
2801
Regards,
Dragon.Jade :-)
