Write the three digit number as 100a+10b+c, where a, b and c are the digits. Write 1002 as 1000+2, now expand (100a+10b+c)(1000+2)=
100000a+10000b+1000c+200a+20b+2c=100000A+10000B+70+C.
I found it easier to equate digits by comparison by arranging this sum as a layout in long multiplication:
............................................1 0 0 2
...............................................a b c
...........................................c 0 0 2c
........................................b 0 0 2b 0
.....................................a 0 0 2a 0 0
.....................................A B 0 0 7 C
We should be able to equate the digits by position by comparisons. C must be an even number 0, 2, 4, 6 or 8. But in the tens we have 7, so c is 5, 6, 7, 8 or 9, to produce a carryover converting 6 to 7 in the tens. That tells us that b must be 3 or 8 because 2*3+1=7 or 2*8+1=17. But in the hundreds we have 0, so b can't be 8 because there would be a carryover, so b=3. And we can see that 2a must be a number ending in zero, so a=0 or 5. But a can't be zero because we would have a 3-digit number starting with zero making it a 2-digit number so a=5, and 2a=10, which is a carryover to the thousands where c is. But the thousands digit is zero, so c=9.
We have all the digits: a=5, b=3 and c=9 making the number 539. 539*1002=540078.