Question

The six digit number 323a5b6 is divisible by 8 and 9.what is the sum of a and b.

Answer 1

Step-by-step explanation:

First, let’s find the LCM of 8, and 9. Since these are all coprime to each other, their LCM is equal to their product.

So, LCM(8,9) = 72

Since our number has to be divisible by all of these, it must be divisible by their LCM. Thus, 323a5b6 is some multiple of 72

Let’s assume A = B = C = 0 (this is of course not true, but bear with me).

323a5b6

So, we have a remainder of 136 when we divide by 504. So, to get the nearest multiple of 504 above this (we want above and not below because otherwise the number would be in the form 738ABC), all we have to do is add the difference between 504 and 136.

504−136=368

739000+368=739368 , which we know is a multiple of 504. So, one possible solution is:

ABC = 368

However, we’re not quite done. Any number that’s divisible by 504 is divisible by 7, 8, and 9. So, if we add 504 to this number, it will still satisfy our requirements.

739368+504=739872 . So the other possible solution is:

ABC = 872

Since adding 504 to this will increase the thousands, these are the only two possibilities