Online Number Properties #5

[Carla]

Just in case I posted this in the wrong section before I wanted to post this question here:


My question is related to the following
(#5 from the online Number Properties question bank)

"When the positive integer x is divided by 11, the quotient is y and the remainder 3. When x is divided by 19, the remainder is also 3. What is the remainder when y is divided by 19?


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I was able to establish two equation: x=11y +3, and x=19z+3. - From there I was not sure what to do and so I had guessed the answer (incorrectly).


The given solution equated both of the equations I had come up with to obtain 11y = 19z

The soltion then goes on to state the following:

"The question asks us what the remainder is when y is divided by 19. From the equation we see that 11y is a multiple of 19 because z is an integer. y itself must be a multiple of 19 since 11, the coefficient of y, is not a multiple of 19.

If y is a multiple of 19, the remainder must be zero."

I do not understand this - why does y itself have to be a multiple of 19?

Thanks

[StaceyKoprince]

Hi, Carla - this one can go in the regular math section since it is not from a practice test (we're just trying to segregate out the practice test questions so that people who don't want to be "spoiled" can easily avoid seeing test questions).

Not quite sure from your post below - did you also post this in the math section? If not, just go ahead and paste it over there and we'll answer it!