Question

Does this congruence hold:
12,345,678,987,654,321 ≡ 0 (mod12,345,678)

Answer 1

I know it means it needs to be perfectly divisible, but is there anyway to figure out if it is without a calculator?

Answer 2

Is this a real problem? Or you made it up? :)

Answer 3

Real problem

Answer 4

you can reduce it to:

987,654,321

Answer 5

So, it would be an integer, thus congruent, correct?

Answer 6

n ≡ 0 (mod12,345,678)

doesn't this require n to be even?

Answer 7

The numerator is odd and the denominator is even how could it hold?

Answer 8

I'm not familiar with divisibility rules

Answer 9

no sorry @kt4rest what i meant was you could reduce the problem to:

987,654,321 ≡ 0 (mod12,345,678)

since

12,345,678,000,000,000 is a multiple of the base

but an odd number cannot be a multiple of an even number, as FFM said

Answer 10

The next problem in the book is asking if the congruence holds for:
12,345,678,987,654,321 ≡ 0 (mod12,345,679)
So you may be right

Answer 11

That sounds correct. So, I would say this is congruent and 12,345,678,987,654,321 ≡ 0 (mod12,345,679) is not?

Answer 12

That holds the quotient is 999999999.

Answer 13

Thank you both