Remainder problems

Question

An integer X, when divided by 4 or 6, results in a remainder of 1. Which of the followimg cannot be a remainder when the same number is divided by 9. 

A 2

B 3

C 7

D 4 

E none 

Answer 1

1. X divided by 4 gives a remainder of 1.  Write this as an equation.   X = 4Y + 1 {Eqn 1}, where Y is any non-negative integer.  For example, when Y =3, then X = 13.  Let's doublecheck:  Does this meet the original statement?  Yes, when 13 is divided by 4, it gives a remainder of 1.
2. X divided by 6 gives a remainder of 1.  Similar to above, X = 6Z + 1 where Z is any non-negative integer.  {Eqn 2}
3. Now we are looking for all the values of X for which both Eqn 1 and Eqn 2 are true.  So, set the two equations equal to one another, i.e. 4Y+ 1 = 6Z +1.  You can subtract the 1 from both sides, and you're left with 4Y = 6Z.  Divide both sides by 4, and you are left with Y = (6/4)Z.  This reduces to Y = 1.5Z
4. Since we need both Y and Z to be integers, this will only be true when Z is an even number.  So we have the following values for Y:  0, 3,  6,  9, 12, ... You see the pattern?
5. Now, we have an equation for X, which is the same as {Eqn 1} but the values for Y are now different, i.e., X = 4Y + 1, and Y = 3n (where n = any non-negative integer)
6. Rewrite by substituting for Y:   X = 4(3n) + 1 =  12n + 1  {Eqn 3}
7. The significance of Eqn 3 is that it gives you all the numbers that satisfy the first part of the problem, which is all the integers which when divided by 4 or 6 will give you a remainder of 1.
8. So, now we can tackle the second part of the problem.  Which number cannot be a remainder when the same number is divided by 9?
9. I think the simplest way to proceed is to plug in integer numbers for n in Eqn 3, beginning with 0:
1. n = 0, X = 1,   Divide-by-9 = 0, Remainder = 1
2. n = 1, X = 13, Divide-by-9 = 1, Remainder = 4
3. n = 2, X = 25, Divide-by-9 = 2, Remainder = 7
4. n = 3, X = 37, Divide-by-9 = 4, Remainder = 1
10. The pattern for the remainder is 1, 4, 7.  This will keep repeating.  The only valid remainders are 1, 4 and 7.
11. So, in terms of the available responses, the options A (2) and B (3) cannot be a remainder.  Now, here's where things get tricky.  The right answer would require you to circle both A and B.  But what if the rules for answering the question require that you can only circle one answer.   Does "which of the following..." mean to select all valid answers?  Or does it mean you can select just one valid answer?  If the former, then you would be correct to circle either A or B.  If the latter, then there is no correct answer among A thru D, and it seems like the answer must be E. 
12. I'm a little surprised at the tricky wording of this problem.  In terms of the math, the answer is that only 1, 4 and 7 are valid remainders.  I leave it up to you, based on the guidelines given for answering the problem, to determine which of the options to select:
1. Circle A and B  (if you are allowed to circle more than one option)
2. Circle A or B (if you believe the wording of the problem allows you to select one of two correct answers)
3. Circle E (if you are not allowed to circle more than one response, and you believe the correct response requires selecting all the options that are not valid remainders.  In effect, you select E because it means A and B)