Number properties questions bank problem:Dividing 11 and 19

[msliwiak]

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?

Answer: 11y=19z

Explanation states that since 11 is NOT a multiple of 19, then y must be a multiple of 19. We know that z is an integer, but how do I eliminate a possibility that y is NOT a multiple of z?

Thank you.
Marek

[venkat_yj]

11y=19z, this can be derived using the problem statements.

For this equation to hold good,

y must be a multiple of 19, as 11 is not. Otherwise the left hand side of the equation cannot be equal to right hand side, which in essence is a multiple of 19.

Likewise, z has to be a multiple of 11, as 19 is not a multiple of 11.

[saurav.raaj]

Hence Remainder = 0?

[venkat_yj]

Yes.

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[Ben Ku]

Hi Marek,

Using the derived relationship 11y = 19z, we know that y = 19z/11. Since y is an integer, then z must be a multiple of 11. Putting it this way:

y = (19)(z/11). So y is a multiple of 19 and z/11. It doesn't really matter whether y is a multiple of z, as long as y is a multiple of z/11.

I hope that makes sense.