There are 900 three-digit numbers starting with 100 and ending with 999. Half of these are odd numbers and half of these are even numbers.

For the number to be odd, it must end in 9. Arranging the other two digits in descending order, the greatest three-digit odd number turns out to be 869.

There are 90 two-digit numbers starting with 10 and ending with 99. Half of these are even numbers and half of these are even numbers.

On arranging the digits in ascending order, the smallest number one gets is 6789 (which is an odd number). On swapping the last two digits, the next number is 6798.

The even numbers are 322 and 456. The odd numbers are 567 and 989. So, the smallest odd number is 567.

Out of the even numbers (678, 768, 786), the greatest number is 786.

2640 > 2604 > 2406 > 2046.

The next even number after 89 is 90. The 12th even number after 90 will be 90 + (12 × 2) = 114.

The 3 ways to express 48 as the product of two even numbers are 2 × 24, 4 × 12, 6 × 8. The fourth way is 3 × 16, but 3 is an odd number.

The 103rd even number after 118 will be 118 + (103 × 2) = 324.

The next odd number after 192 is 193. The 13th odd number after 193 will be 193 + (13 × 2) = 219.

The next odd number after 292 is 293. The 105th odd number after 293 will be 293 + (105 × 2) = 503.

When a number is multiplied by 5, it gives a number with a 0 or 5 in the units place. Only an odd number when multiplied by 5 gives an odd number with a 5 in the units place.

Only an odd number when divided by 2 leaves 1 as the remainder.

The sum of two odd numbers will always be an even number. If an even number is divided by 2, the remainder is 0.

The third odd number serially after 197 is 203. Note that 197 + 6 = 203.

The fourth odd number serially after 198 is 205. Note that 198 + 7 = 205.

9 + 5 + 7 + 4 = 25. s s s 8 + 3 + 1 + 3 = 15.
7 + 6 + 8 + 5 = 26. s s s 6 + 9 + 2 + 4 = 21.
The sum of the digits in 7685 is 26 (an even number).

The odd numbers between 30 and 40 are 31, 33, 35, 37, 39. Their sum is (30 × 5) + (1 + 3 + 5 + 7 + 9) = 150 + 25 = 175. Note that 175 is simply 5 times the middle number (i.e., 5 × 35 = 175).

The even numbers between 41 and 51 are 42, 44, 46, 48, 50. Their sum is (40 × 5) + (2 + 4 + 6 + 8 + 10) = 200 + 30 = 230. Note that 230 is simply 5 times the middle number (i.e., 5 × 46 = 230).

The sixth, seventh and eighth even numbers after 70 are 82, 84, 86. Their sum is (80 × 3) + (2 + 4 + 6) = 240 + 12 = 252.

The sixth, seventh, eighth and ninth odd numbers after 30 are 41, 43, 45, 47. Their sum is (40 × 4) + (1 + 3 + 5 + 7) = 160 + 16 = 176.

The difference between consecutive odd numbers is 2. Therefore the difference between the 23rd and the 26th odd numbers will be 2 × (26 − 23) = 6.

The difference between consecutive even numbers is 2. Therefore the difference between the 17th and the 21st even numbers will be 2 × (21 − 17) = 8.

The even numbers are 22, 24, 26, 28, 30. The odd numbers are 21, 23, 25, 27, 29. Thus, the answer is
(22 + 24 + 26 + 28 + 30) − (21 + 23 + 25 + 27 + 29)
= (22 − 21) + (24 − 23) + (26 − 25) + (28 − 27) + (30 − 29)
= 1 + 1 + 1 + 1 + 1 = 5.

The greatest four-digit odd number is 9999. The greatest three-digit even number is 998. Their sum is 10997.

The greatest four-digit even number is 9998. The smallest four-digit odd number is 1001. Their difference is 8997.

The 30th even number after 26 is 26 + (2 × 30). The 50th even number after 26 is 26 + (2 × 50). The sum of these gives
(2 × 26) + (2 × 30) + (2 × 50)
= 2 (26 + 30 + 50) = 2 × 106 = 212.

The 2 three-digit odd numbers that can be formed are 507 and 705.

The 4 three-digit even numbers that can be formed are 236, 326, 362 and 632.