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

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swerve: Legendary Member; Posts: 2499; Joined: Sun Oct 29, 2017 2:04 pm; Followed by:6 members

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

by swerve » Wed Jul 04, 2018 3:10 pm

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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

The OA is C.

Please, can anyone explain this PS question? I don't understand why C is the correct answer. I need help. Thanks.

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Source: Beat The GMAT — Problem Solving | Wed Jul 04, 2018 3:10 pm

Shahrukh_mbabreakspace: Senior | Next Rank: 100 Posts; Posts: 32; Joined: Mon Jul 02, 2018 1:56 am; Followed by:2 members

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by Shahrukh_mbabreakspace » Wed Jul 04, 2018 10:21 pm

P leaves 1 as remainder when divided by 9, it means p is of 9x+1 form where x is a non-negative integer
It can be odd or even depending on value of x, so I and II can't be true always.

We can write P as 3*3x +1
This clearly Shows that III is always true and Z is of 3x form.

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Last edited by Shahrukh_mbabreakspace on Thu Jul 05, 2018 2:11 am, edited 1 time in total.

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GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pm; Location: New York, NY; Thanked: 13060 times; Followed by:1906 members; GMAT Score:790

by GMATGuruNY » Thu Jul 05, 2018 2:06 am

swerve 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 dividing p by 9 yields a remainder of 1, p is ONE MORE THAN MULTIPLE OF 9:
p = 9a + 1, where a is a nonnegative integer.
Options for p
1, 10, 19, 28...

If p=1, then Statement I is not true.
Eliminate A, D and E.
If p=10, then Statement II is not true.
Eliminate B.

The correct answer is C.

The expression in blue indicates that Statement III must be true.

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Scott@TargetTestPrep: GMAT Instructor; Posts: 8086; Joined: Sat Apr 25, 2015 10:56 am; Location: Los Angeles, CA; Thanked: 43 times; Followed by:29 members

by Scott@TargetTestPrep » Sat Jul 07, 2018 5:31 pm

swerve 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

We see that p can be values such as 1, 10, 19, 28, ....

Therefore, neither Roman numeral I nor Roman numeral II is true. Furthermore, there is no answer choice stating that none of the Roman numerals is true.

Thus, the Roman numeral that must be true is that p = 3z+1 for some integer z.

Alternate Solution:

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

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