Q: Product of 3 consecutive even integers is 87,***,**8 where * are missing digits. Find missing digits without a calculator?
I know the answer is 87,526,608 with the even integers 442, 444, 446 but that is with a calculator. I should find it without a calculator.
I found the ones and hundreds digit of the even integer easily. The ones digit can be 0, 2, 4, 6, 8. We try the product of 3 consecutive even digits and only the product of 2, 4, and 6 have 8 as ones. So, the consecutive even integers have to be **2, **4, **6.
Next for the hundreds I checked 400x400x400 and it is 64,000,000 while 500x500x500 is 125,000,000. 87,***,**8 is between these numbers so the hundreds digit of the middle number has to be 4. Since the ones are 2, 4, and 6, the hundreds digit of all three consecutive even integers is 4.
Now how should I proceed to get the tens digit OR is there a way to get all the missing digits without knowing the 3 consecutive even digits.
Edit: Ok I am getting the exact same answers that involve cubing of 2-digit numbers. This is not allowed.