I'm currently studying for the GRE and a specific type of question has me stumped. I have a two part question, but first here is one problem and my work:
"If x is the remainder when a multiple of 4 is divided by 6, and y is the remainder when a multiple of 2 is divided by 3, what is the greatest possible value of x+y"
Since it says greatest I thought GCM and wrote(with arbitrary variables)
$$N = 24a + x$$
$$M = 6b + y$$
I thought that since 2 & 3 are factors of 4 & 6, respectively, maybe I could equate them:
$$24a+x=6b+y$$
Or maybe N is twice a large as M:
$$24a+x=2(6b+y)$$
1) How do I solve this? Am I even on the right track?
This kind of question has had me stumped for a while but the books explanation has me even more lost.
"The greatest possible remainder for a multiple of 4 being divided by 6, happens when 4 is divided by 6. When 4 is divided by 6 the result is 0 with a remainder of 6."
…..Don't remainders by nature have to be smaller than the divisor……
2)How is that possible? Shouldn't the result be 0 with a remainder of 4?
How can there be a remainder of 6 when the divisor is 6? I found a similar problem and solution here: Finding a number given its remainder when divided by other numbers and here: How to find the greatest remainder of a number that is a multiple of another number But the more I read the more lost I was.
At this point I feel like a child throwing spaghetti about trying to make art. I have no idea how to even start on this problem any help is appreciated.
Best Answer
When any integer $a$ is divided by $6$ then the possible remainders are $0,1,2,3,4,5$. So if $4a$ is divided by $6$, then the possible remainders will be $0,4,2,0,4,2$ respectively (if you are familiar with modular arithmetic then this can be written more elegantly). So the maximum possible remainder when a multiple of $4$ is divided by $6$ is $4$.
Likewise when you divide any integer $b$ by $3$, the possible remainders are $0,1,2$. In which case the remainders when $2b$ is divided by $3$ will be $0,2,1$. Here the maximum is $2$. Thus the maximum of the sum will be $4+2=6$.