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# The integer P is greater than 7. If the integer P leaves a remainder

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Math Expert
Joined: 02 Sep 2009
Posts: 42305

Kudos [?]: 133076 [0], given: 12403

The integer P is greater than 7. If the integer P leaves a remainder [#permalink]

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12 Oct 2017, 23:51
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68% (01:25) correct 32% (01:56) wrong based on 109 sessions

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The integer P is greater than 7. If the integer P leaves a remainder of 4 when divided by 9, all of the following must be true EXCEPT

A. The number that is 4 less than P is a multiple of 9.
B. The number that is 5 more than P is a multiple of 9.
C. The number that is 2 more than P is a multiple of 3.
D. When divided by 3, P will leave a remainder of 1.
E. When divided by 2, P will leave remainder of 1.
[Reveal] Spoiler: OA

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Kudos [?]: 133076 [0], given: 12403

Intern
Joined: 15 Aug 2016
Posts: 39

Kudos [?]: 1 [0], given: 55

The integer P is greater than 7. If the integer P leaves a remainder [#permalink]

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13 Oct 2017, 04:20
We can Make different cases in this question.
For example lets take P which satisfies the conditions given in the questions i.e multiple of 9 plus 4.
So it can be 31,40,49
a)Always True (27,36,45 are multiples of 9)
b)Always true(Negative remainder , so this will also be true)
c)Always true( 33,42,51 are multiples of 3)
d)Always true( Leaving remainder 1) (Also remainder 4 in the question divide by 3 also leaves 1 as remainder)
e)Not always True. (40 is divisible by 2)

Thus OA=E.
Please correct me if I am wrong.

Kudos [?]: 1 [0], given: 55

Director
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The integer P is greater than 7. If the integer P leaves a remainder [#permalink]

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15 Oct 2017, 07:23
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Bunuel wrote:
The integer P is greater than 7. If the integer P leaves a remainder of 4 when divided by 9, all of the following must be true EXCEPT

$$P = 9k+4$$, where $$k$$ is any positive integer and since $$P>7$$ so $$k$$ is not equal to $$0$$

A. The number that is 4 less than P is a multiple of 9. $$=>9k+4-4=9k$$. clearly a multiple of $$9$$. True

B. The number that is 5 more than P is a multiple of 9. $$=>9k+4+5=9k+9$$. clearly a multiple of $$9$$. True

C. The number that is 2 more than P is a multiple of 3. $$=>9k+4+2=9k+6=3(3k+2)$$. clearly a multiple of $$3$$. True

D. When divided by 3, P will leave a remainder of 1. $$=>\frac{9k+4}{3}=3k+\frac{4}{3}$$. This will leave a remainder $$1$$. True

E. When divided by 2, P will leave remainder of 1. $$=>9k+4$$, if $$k$$ is even then it will be divisible by $$2$$ but if $$k$$ is odd then it will leave a remainder of $$1$$ when divided by $$2$$. Hence Not True always

Option E

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Re: The integer P is greater than 7. If the integer P leaves a remainder [#permalink]

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16 Oct 2017, 17:12
Bunuel wrote:
The integer P is greater than 7. If the integer P leaves a remainder of 4 when divided by 9, all of the following must be true EXCEPT

A. The number that is 4 less than P is a multiple of 9.
B. The number that is 5 more than P is a multiple of 9.
C. The number that is 2 more than P is a multiple of 3.
D. When divided by 3, P will leave a remainder of 1.
E. When divided by 2, P will leave remainder of 1.

We can create the following equation:

P = 9Q + 4

Thus, we see that P can be values such as:

13, 22, 31, 40, ...

A) The number that is 4 less than P is a multiple of 9.

P - 4 = 9Q + 4 - 4

P - 4 = 9Q

A is true.

B) The number that is 5 more than P is a multiple of 9.

P + 5 = 9Q + 4 + 5

P + 5 = 9Q + 9

B is true.

C) The number that is 2 more than P is a multiple of 3.

P + 2 = 9Q + 4 + 2

P + 2 = 9Q + 6

C is true.

D) When divided by 3, P will leave a remainder of 1.

(9Q + 4)/3 = (3Q + 1) + 1/3

D is true.

E) When divided by 2, P will leave remainder of 1.

(9Q + 2)/2 = (4Q + 1) + Q/2

We can’t be certain what the remainder is. If Q = 1, then the remainder is 1; however, if Q is 2, then the remainder is 0.

Alternate solution:

Since the problem says all of the answer choices must be true except one of them, we can use any integer > 7 that satisfies the condition “when it is divided by 9, it will leave a remainder of 4” to check the answer choices. Since 13 is one such number, we will use that.

A) 13 - 4 = 9 and 9 is a multiple of 9. This is true.

B) 13 + 5 = 18 and 18 is a multiple of 9. This is true.

C) 13 + 2 = 15 and 15 is a multiple of 3. This is true.

D) 13/3 = 4 R 1. This is true.

E) 13/2 = 6 R 1. This is true.

Since 13 makes all answer choices true, we need to use another number. Another number we can use is 13 + 9 = 22.

Instead of checking each answer choice again, we can see that choice E is not true, since when 22 is divided by 2, the remainder is 0, not 1.

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Jeffery Miller

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Re: The integer P is greater than 7. If the integer P leaves a remainder   [#permalink] 16 Oct 2017, 17:12
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