If p and q are positive integers. If p is divided by 2, the remainder

Question

If p and q are positive integers. If p is divided by 2, the remainder is 1; and if q is divided by 6, the remainder is 1. Which of the following must be true.

I. pq + 1 is even.
II. pq/2 is an integer.
III. pq is a multiple of 12.

(A) I only
(B) II only
(C) III only
(D) I and II only
(E) I and III only

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findingmyself · Answer

I. pq + 1 is even. 
 II. pq/2 is an integer. 
 III. pq is a multiple of 12.

We are asked for MUSTBE true. Always beware of MUST BE and COULD BE, While MUST BE says the answer has to be correct in all conditions within the constraints give. Let's first call out the constrains

P and Q should be positive: 1,2,3,4, 
 P and Q CAN BE same value, since question is not saying they are different

Let's express P and Q as two equations 
 P=2a+1, where a is the multiplier 
 Q=6b+1, where b is the multiplier

I. PQ+1 is even, which is (2a+1)(6b+1)+1=12ab+2a+6b+2= Even+Even+Even+Even=EVEN. This will be even, in cases where a and b are zero. (Always check for this). a and b are zero means P=Q=1, in this case, still the value will be even= MUST be true

II. PQ/2= 12ab+2a+6b+1, this cannot be true when the value of P and Q is 1, (1*1=1,1/2=0.5) not an Integer, Could be true, but not a must be true condition

III. PQ is a multiple of 12, this cannot be true in all cases, consider the same reason as in II

Souradeep17 · Answer

From the given condition we can confirm p is odd 
 And q is also odd as a multiple of 6 is always even and +/-1 to that will result in an odd number.

Now let's take two number 5 and 13

For option 1 5*13+1=66 yes it is possible.

Also multiplication of two odd numbers results in odd and addition of 1 to it results in even number so option 1 possible.

Option two 5*13/2 is not even not possible.

Option 3 5*13 is 65 which is not multiple of 12 hence this is also not true .
So option answer should option A 1 only

GraemeGmatPanda · Answer

We translate "p is divided by 2, the remainder is 1" by representing p as an odd integer.
 p = 2k + 1

We translate "q is divided by 6, the remainder is 1" by writing q in terms of an integer multiple of 6 plus 1.
 q = 6m + 1

We form the product and expand to analyze each statement.
 pq = (2k + 1)(6m + 1) = 12km + 2k + 6m + 1

To test Statement I, we add 1 and factor out 2 to show evenness.
 pq + 1 = 12km + 2k + 6m + 1 + 1 = 12km + 2k + 6m + 2 = 2(6km + k + 3m + 1)

To test Statement II, we check if pq is divisible by 2 by writing it in the form  2(	ext{integer}) + 1 .
 pq = 12km + 2k + 6m + 1 = 2(6km + k + 3m) + 1

To test Statement III, we note that only the term  12km  is a multiple of 12, leaving  2k + 6m + 1  which in general is not a multiple of 12.
 pq = 12km + (2k + 6m + 1)

Answer A

Hope this helps! 
ʕ•ᴥ•ʔ

If p and q are positive integers. If p is divided by 2, the remainder

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If p and q are positive integers. If p is divided by 2, the remainder

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