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How many ordered pairs of positive integers (M,N) satisfy the equation

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Bunuel
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How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   22 Apr 2019, 01:46
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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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D
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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   22 Apr 2019, 03:30
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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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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   23 Apr 2019, 17:02
LCM of 36 = 9

Each factor will always be multiplied by another number.

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

D
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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   26 Apr 2019, 15:33
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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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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   31 Mar 2020, 05:42
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


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

M/6 = 6/N
MN = 36 ; N <>0

36 = 2^2 * 3^2

(M,N) = {(1,36),(2,18),(3,12),(4,9),(6,6),(9,4),(12,3),(18,2),(36,1)}

9 ordered pairs of M & N satisfy the equation

IMO D
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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink] New post   10 Feb 2022, 11:08
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Re: How many ordered pairs of positive integers (M,N) satisfy the equation [#permalink]
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