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Math Expert V
Joined: 02 Sep 2009
Posts: 57200
How many ordered pairs of positive integers (M,N) satisfy the equation  [#permalink]

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Difficulty:   45% (medium)

Question Stats: 52% (01:18) correct 48% (01:33) wrong based on 48 sessions

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How many ordered pairs of positive integers (M,N) satisfy the equation $$\frac {M}{6} = \frac{6}{N}$$ ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

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e-GMAT Representative V
Joined: 04 Jan 2015
Posts: 3018
Re: How many ordered pairs of positive integers (M,N) satisfy the equation  [#permalink]

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Solution

Given:
• M and N are positive integers satisfying the equation M/6 = 6/N

To find:
• The number of ordered pairs satisfying the equation

Approach and Working:
• M/6 = 6/N
Or, MN = 36

As both M and N are positive integers, the number of ways we can write MN = 36 as follows:
• 1 x 36
• 2 x 18
• 3 x 12
• 4 x 9
• 6 x 6
• 9 x 4
• 12 x 3
• 18 x 2
• 36 x 1

Hence, the correct answer is option D.

Answer: D

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Senior Manager  S
Joined: 12 Sep 2017
Posts: 297
Re: How many ordered pairs of positive integers (M,N) satisfy the equation  [#permalink]

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LCM of 36 = 9

Each factor will always be multiplied by another number.

Ex. 36 - a factor of 36.... 36*1 = 36

D
Target Test Prep Representative D
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 7420
Location: United States (CA)
Re: How many ordered pairs of positive integers (M,N) satisfy the equation  [#permalink]

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1
Bunuel wrote:
How many ordered pairs of positive integers (M,N) satisfy the equation $$\frac {M}{6} = \frac{6}{N}$$ ?

(A) 6
(B) 7
(C) 8
(D) 9
(E) 10

Cross-multiplying, we see that MN = 36. Notice that if k is a positive factor of 36, then k x (36/k) = 36; therefore (k, 36/k) is one of the ordered pairs we are looking for. Since this is true for any positive factor of 36, the number of such ordered pairs is precisely the number of positive factors of 36.

Since 36 is a perfect square, we see that it has an odd number of factors, and thus the number of ordered pairs must also be odd, so the correct answer is either 7 or 9.

Since 36 = 2^2 x 3^2, it has (2 + 1)(2 + 1) = 9 factors and thus there are 9 ordered pairs (M, N) satisfying the equation.

Answer: D
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If you find one of my posts helpful, please take a moment to click on the "Kudos" button. Re: How many ordered pairs of positive integers (M,N) satisfy the equation   [#permalink] 26 Apr 2019, 15:33
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# How many ordered pairs of positive integers (M,N) satisfy the equation

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