If positive integer p divided by 9 leaves a remainder of 1, which of

Question

If positive integer p divided by 9 leaves a remainder of 1, which of the following must be true?

I. p is even.
II. p is odd.
III. p = 3z + 1 for some integer z.

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

Source:   Nova GMAT  
Difficulty Level:   500

sashiim20 · Answer

The number is in the form,  p = 9x + 1  --------- (x is the quotient )
Hence number when divided by 3 will leave remainder 1. 
Therefore number must be in the form;  p = 3z + 1  ---------- (z is the quotient) (III)

Lets try even and odd numbers for p.
If p is 10, divided by 9 will leave remainder 1.
If p is 19 , divided by 9 will leave remainder 1.

Therefore p could be even or odd. Therefore I and II must not be true.

Only III.  Answer C...

Abhishek009 · Answer

Values of p can be 1 , 10 , 19 , 28.........

I. p is even   (Can be both even/odd)  
II. p is odd   (Can be both even/odd)  
 III. p = 3z + 1 for some integer z -  True in all possible cases...

Thus, answer will (C)

generis · Answer

Try to disprove each statement.

\frac{p}{9}  = a + 1, so p = 9a + 1

p = 10, 19, 28, 37, 46, 55 ... (leaving out 1 on purpose)

From list, neither I nor II is true. 19, odd, and 10, even, disprove each respectively. At this point you could choose (C) and move on, because all other answers contain I and/or II

III. p = 3z + 1 for some integer z

10 = 3 (3)  + 1. 
19 = 3 (6)  + 1. 
55 = 3 (18)  + 1. 
III is true. Answer C

ScottTargetTestPrep · Answer

Since positive integer p divided by 9 leaves a remainder of 1, we have:
 
p/9 = Q + 1/9
 
p = 9Q + 1
 
Thus, p can be values such as 1, 10, 19, 28, ….
 
Thus, p can be either even or odd.
 
Now we can look at Roman numeral III:
 
p = 3z + 1 for some integer z.
 
Recall that we have expressed that p = 9Q + 1, so p = 3(3Q) + 1. We see that if we let z = 3Q, we can express p as 3z + 1. So Roman numeral III is true. 
 
Answer: C

sonikavadhera · Answer

P CAN BE 1,10,19....

hence I and II can be true or false

III is true for all cases

Option C

FB2017 · Answer

p=3z+1 should be true where 1 is the remainder as per problem statement.
Since it is divisible by 9 z is quotient and the above expression cabe written as multiple of 3.
Option C.

bumpbot · Answer

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If positive integer p divided by 9 leaves a remainder of 1, which of

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