A number theory problem by Akshat Sharda

We can approach by complementary counting. The sum of numbers containing only odd numbers is obtained to be 69375 & the sum of the numbers containing only even numbers is obtained as 54400(55500-1100) Then we subtract the above results from the total sum of all 3-digit numbers(494550) to obtain 370775 as the answer...

All 3-digit natural numbers with at least one odd digit and one even digit means all 3-digit natural numbers except those with 3 odd digits or 3 even digits.

The required sum $S_{oe*}$ $=$ the sum of all 3-digit numbers $S_{***}$ $-$ the sum of all 3-odd digit numbers $S_{ooo}$ $-$ the sum of all 3-even digit numbers $S_{eee}$ or $S_{oe*} = S_{***} - S_{ooo} - S_{eee}$ .

Three-digit numbers range from $a=100$ to $l=999$ , altogether $n=900$ numbers. Therefore,

$\begin{aligned} S_{***} = \frac {n(a+l)}2 = \dfrac {900(100+999)}2 = 494550 \end{aligned}$

Since there are 5 odd digits (1, 3, 5, 7, 9), the number of 3-odd digit numbers $n_{ooo} = 5\times 5 \times 5 = 125$ . We note the each odd digit appear the same number of times as the unit, tenth and hundredth digit, that is $125/5= 25$ times. Therefore,

$\begin{aligned} S_{ooo} & = (1+3+5+7+9)\times 25 \times 100 + (1+3+5+7+9)\times 25 \times 10 + (1+3+5+7+9)\times 25 \times 1 \\ & = 25 \times 25 \times 111 = 69375 \end{aligned}$

For 3-even digit numbers, however, starts with 200 to 888, and $n_{eee} = 4 \times 5 \times 5 = 100$ . Out of these 100 numbers, (2, 4, 6, 8) share as the 100 hundredth digit, that is $100/4 = 25$ , but (0, 2, 4, 6, 8) share as the tenth and unit digits, that is $100/5=20$ . Then, we have

$\begin{aligned} S_{eee} & = (2+4+6+8)\times 25 \times 100 + (0+2+4+6+8)\times 20 \times 10 + (0+2+4+6+8)\times 20 \times 1 \\ & = 20 (2500+200+20) = 54400 \end{aligned}$

Therefore, $S_{oe*} = S_{***} - S_{ooo} - S_{eee} = 494550 - 69375 - 54400 = \boxed{370775}$ .

This question is indeed very very long,

First of all we sum all 3-digit numbers, S=450(1099) S=494550

Now we identify those numbers whose sum is not required and add them,

111,113,...,119 => 675

131,133,...,139 => 775

And so on,

As a whole the sum comes out to be 123775,

Now we subtract this from total,

494550-123775=370775