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For single digit numbers in the range 0-9 there are 4 (excludes zero).

For 2 digit numbers in the range 10-99, then the even sum condition only applies when both digits are even, i.e. such numbers ending in 0,2,4,6,8 in the 20s, 40s, 60s & 80s. There are 4 x 5 = 20.

For 3 digit numbers in the range 100-999 then the even sum condition applies when (1) All 3 digits are even or (2) The first 2 digits are odd and the third digit is even.

(1) There are 4 options for the 1st digit (2,4,6,8), and 5 options for the 2nd and 3rd digits (0,2,4,6,8) There are thus 4 x 5 x 5 = 100 qualifying numbers.

(2) If the 1st and 2nd digits are odd and the 3rd digit is even then there are 5 x 5 x 5 = 125 qualifying numbers.

Total number of qualifying numbers = 4 + 20 + 100 + 125 = 249

Q: How many positive even integers up to 1000 inclusive have an even digit sum?

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There are 1,000 positive integers between 1,000 and 9,999, inclusive, that are divisible by nine.

There are 3168 such numbers.

750

Infinitely many, since every positive number is a square of some other number. However, there are 21 perfect squares (SQUARES OF INTEGERS). From 112 to 312, inclusive.

332

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The sum of the positive integers from 1,000 to 1,100 inclusive is: 106,050

There are 1,000 positive integers between 1,000 and 9,999, inclusive, that are divisible by nine.

There are 3168 such numbers.

There are 200 positive four digit integers that have 1 as their first digit and 2 or 5 as their last digit. There are 9000 positive four digit numbers, 1000 through 9999. 1000 of them have 1 as the first digit, 1000 through 1999. 200 of them have 2 or 5 as their last digit, 1002, 1005, 1012, 1015, ... 1992, and 1995.

All multiples of 3 have digits that add up to a multiple of 3. There are 333 multiples of 3 between 1 and 1000.

from 1000 to 9999, ie 9999 - 999 = 9000

All multiples of 3 have digits that add up to a multiple of 3. There are 333 multiples of 3 between 1 and 1000.

The sum of the squares of the first 1000 positive odd integers (from 12 to 19992) is 1333333000.

From 1000 to 9999 (inclusive) there are 9999 - 1000 + 1 = 9000 integers. Half of which are even, half of which are odd. So the answer is 9000/2 = 4,500.

750

666 integers.

1,2,3,4,5,6,7,8,9,10 all the way thru 1000.