The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

This problem trains for: SAT-I, AMC-8, GMAT, AMC-10.

Let n be a natural number different from zero. Dividing it by 19 we get a remainder of 5. Dividing it by 22 we get a remainder of 6. Is the quotient resulting from the division by 18 odd or even? Is it divisible by 11?

Using the theorem of integer division with remainder we can write n as:

equation

and as:

equation

Therefore,

equation

equation

We can identify the quotient of the division of n by 19 as the number denoted by m.

Due to the uniqueness of the prime factorization of a number, m must be a multiple of 11. Also, since the right hand side is odd:

equation

the left hand side must also be odd and therefore m must be odd as well.

m is odd and divisible by 11.