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

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If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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15 Jun 2017, 05:15
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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

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If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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15 Jun 2017, 05:35
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Bunuel wrote:
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

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.

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Re: If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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15 Jun 2017, 08:03
Bunuel wrote:
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

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...

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If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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15 Jun 2017, 08:34
Bunuel wrote:
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

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.
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Re: If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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17 Jun 2017, 05:09
Bunuel wrote:
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

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.

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Re: If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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21 Jun 2017, 02:07
P CAN BE 1,10,19....

hence I and II can be true or false

III is true for all cases

Option C
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Re: If positive integer p divided by 9 leaves a remainder of 1, which of  [#permalink]

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16 Aug 2017, 08:57
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.
Re: If positive integer p divided by 9 leaves a remainder of 1, which of   [#permalink] 16 Aug 2017, 08:57
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