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OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

Real problem

OpenStudy (anonymous):

you can reduce it to: 987,654,321

OpenStudy (anonymous):

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

OpenStudy (anonymous):

n ≡ 0 (mod12,345,678) doesn't this require n to be even?

OpenStudy (anonymous):

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

OpenStudy (anonymous):

I'm not familiar with divisibility rules

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

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

OpenStudy (anonymous):

That holds the quotient is 999999999.

OpenStudy (anonymous):

Thank you both

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