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# For the positive integers q, r, s, and t, the remainder when

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Manager
Joined: 06 Jun 2012
Posts: 82
Concentration: Technology, Entrepreneurship
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For the positive integers q, r, s, and t, the remainder when  [#permalink]

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21 Jul 2012, 12:38
4
29
00:00

Difficulty:

65% (hard)

Question Stats:

59% (02:15) correct 41% (02:33) wrong based on 329 sessions

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For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

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Re: For the positive integers q, r, s, and t the remainder when  [#permalink]

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21 Jul 2012, 13:09
3
8
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

1)32
2)38
3)44
4)52
5)63

Its obvious from the prompt that r is greater than 7 and t is greater than 3
so the product of rt must have at least 2 factors, one which is greater than 7 and other greater than 3
Now lets break down the possible factors so that this condition is validated
The (r,t) pairs for the options can be:
for 32, (8,4)
for 44, (11,4)
for 52 (13,4)
for 63 (9,7)
however 32 can only be written as 38*1 or 19*2 none of which can validate the prompt.
Hence answer is B
Hope this helps
Cheers
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Manager
Joined: 22 Dec 2011
Posts: 231
For the positive integers q, r, s, and t, the remainder when  [#permalink]

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24 Oct 2012, 21:57
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT

A. 32
B. 38
C. 44
D. 52
E. 63

From the question we know that r >= 8 and s >=4

So A is possible, from there how to proceed?

Cheers
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Re: For the positive integers q, r, s, and t, the remainder when  [#permalink]

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24 Oct 2012, 22:24
1
Jp27 wrote:
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT

32
38
44
52
63

OA

From the question we know that r >= 8 and s >=4

So A is possible, from there how to proceed?

Cheers

$$32 = 2^5$$
$$38 = 2^1 * 19^1$$
$$44 = 2^2 * 11^1$$
$$52 = 2^2 * 13^1$$
$$63 = 3^2 * 7^1$$
All options other than B can be represented as a product of two numbers one being greater than or equal to 4 and the other being greater than or equal to 8.

So, answer should be B

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Re: For the positive integers q, r, s, and t, the remainder when  [#permalink]

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25 Oct 2012, 03:12
4
1
Jp27 wrote:
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT

A. 32
B. 38
C. 44
D. 52
E. 63

From the question we know that r >= 8 and s >=4

So A is possible, from there how to proceed?

Cheers

Important property: remainder cannot be greater than the divisor.

Therefore, since the remainder when q is divided by r is 7, then r>7;
Similarly, since the remainder when s is divided by t is 3, then t>3.

Now, all answers, except 38 can be represented as the product of two multiples one of which is greater than 7 and another is greater than 3:

32=8*4
44=11*4
52=13*4
63=9*7

However, 38=1*38 or 19*2, thus rt cannot equal to 38.

Hope it's clear.
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Re: For the positive integers q, r, s, and t, the remainder when  [#permalink]

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23 Mar 2015, 01:49
souvik101990 wrote:
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

1)32
2)38
3)44
4)52
5)63

however 32 can only be written as 38*1 or 19*2 none of which can validate the prompt.
Hence answer is B
Hope this helps
Cheers

Hi souvik101990!
I think there is a typo in bolded part of statement. There should be 38 not 32. Yes?
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For the positive integers q, r, s, and t, the remainder when  [#permalink]

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29 Aug 2017, 18:06
TechWithNoExp wrote:
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

r>7
t>3
rt factor can't be<4
38 has two sets of factors: 1*38*and 2*19
1 and 2 are both<4
B
CEO
Joined: 11 Sep 2015
Posts: 3237
Re: For the positive integers q, r, s, and t, the remainder when  [#permalink]

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28 Aug 2018, 12:55
Top Contributor
TechWithNoExp wrote:
For the positive integers q, r, s, and t, the remainder when q is divided by r is 7 and the remainder when s is divided by t is 3. All of the following are possible values for the product rt EXCEPT:

A. 32
B. 38
C. 44
D. 52
E. 63

Great question!!

USEFUL PROPERTY:
When positive integer N is divided by positive integer D, the remainder R is such that 0 ≤ R < D
For example, if we divide some positive integer by 7, the remainder will be 6, 5, 4, 3, 2, 1, or 0
Conversely, if I know that, when k is divided by w, the remainder is 5, then I know that w must be greater than 5

The remainder when q is divided by r is 7
This tells us that r is greater than 7

s is divided by t is 3
This tells us that t is greater than 3

Now check the answer choices...

A) 32
Is it POSSIBLE for rt to equal 32?
Yes, if r = 8 and t = 4, then rt = 32
ELIMINATE A

B) 38
Is it POSSIBLE for rt to equal 38?
NO.
There are only two ways to write 38 as the product of POSITIVE INTEGERS:
i) (2)(19) = 38
ii) (1)(38) = 38
If r is greater than 7 and t is greater than 3, there's no way that one of the values (r or t) can equal 1 or 2.

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Re: For the positive integers q, r, s, and t, the remainder when &nbs [#permalink] 28 Aug 2018, 12:55
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# For the positive integers q, r, s, and t, the remainder when

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