Say no to calculator!

If you are attempting to find the last 3 digits - only the last three digits of each factor is required
So 789 X 321 =

1 X 789 = 789
20X 789 = 15,780 ( the 15,000 is irrelevant)
300 X 789 = 236,700 ( the 236,000 is irrelevant)

So - add the following 789 + 780 + 700 = 2269 - last three digits are 269

This is Very simple, And it can be approached with Modular Arithmetic

We get $123456789 = 789\mod1000$ .

Again $987654321 = 321\mod1000$ .

Multiplying these two gives us

$123456789 * 987654321 = 269 \mod1000$

So $\boxed{269}$ are the last three digits of the number.

To evalute the last three digits of $123456789 \times 987654321$ we just need to focus on the 3-digit numbers $789\times 321 = 253269$ last digits are $\boxed{269}$

Further more

The difference is $789-321 = 468$ . Dividing $468$ by $2$ we get $234$ and $789\times 321$ can be written as

$(555 + 234)\times (555-234)$

$= (555)^2 - (234)^2$

$= 308025 - 54756$

We require 3-digit number however $025$ is smaller than $756$ so better is to 4digit number 1025 which is nearest to $756$ (which is not compulsory as we can take 2025,3025…8025)

$= 1025 - 756 = 269$

So last three digit are $\boxed{269}$ .

In this case we only need to consider the product of last three digits of the two numbers $789 \times 321$ because the rests of the numbers will not affect the last three digits of the product. Consider the long multiplication below.

$\begin{array} {lr} & 789 \\ \times & 321 \\ \hline & 789 \\ & 1578 \ \ \\ & 2367 \ \ \ \ \\ \hline & \cdots 269 \\ \hline \end{array}$

Therefore, the last three digits of the product is $\boxed{289}$ .

Alternatively, using modular arithmetic, we have:

$\begin{aligned} 789 \times 321 & \equiv 789 \times 1 + 89 \times 20 + 9 \times 300 \text{ (mod 1000)} \\ & \equiv 789 + 1780 + 2700 \text{ (mod 1000)} \\ & \equiv 789 + 780 + 700 \text{ (mod 1000)} \\ & \equiv 2269 \equiv \boxed{269} \text{ (mod 1000)} \end{aligned}$

$123456789 \times 987654321 \equiv 789 \times 321 \equiv -211 \times 321 \equiv -731 \equiv 269 \pmod{1000}$

Hence, the answer is $\boxed{269}$ .

Guys it's​just simple trick. Remember how u multiply in lower classes So in that way 123456789 987654321

.............789 .............78 .............7

..............269

You can't use a calculator. How I solved it is multiplying the last three digits 9 x 1 = 9(last digit) 8 x 2 = 16(don't write 16 only 6 and on the next one will add the 1)(middle digit) 7 x 3 = 21 + 1 = 22(don't write 22 only 2 and on the next one will add the 2) and get the answer 269.

Vedic method for multiplication saved my day.

Other solutions there are enough to explain in an easy way. Here is the traditional one

$123456789 \times 987654321 \\ = \dfrac{81}{81} \times 123456789 \times 987654321 \\ = \dfrac{123456789 \times (987654321 \times 81)}{81} \\ = \dfrac{123456789 \times 80000000001}{81} \\ = 121932631112635269$