6 up and 6 down

How many ordered pairs of positive integers ( M , N ) ({M},{N}) , satisfy the equation below?

M 6 = 6 N \large \dfrac{{M}}{6}= \dfrac{6}{{N}}

9 6 \infty 7 8

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2 solutions

Anish Harsha
Feb 25, 2016

M 6 = 6 N \dfrac{\color{#D61F06}{M}}{6}=\dfrac{6}{\color{#3D99F6}{N}}

is a ratio ; therefore, we can cross multiply ,

M N = 36 \color{#D61F06}{M}\color{#3D99F6}{N}=36

Now, we find all the factors of 36,

1 × 36 = 36 2 × 18 = 36 3 × 12 = 36 4 × 9 = 36 6 × 6 = 36 \begin{aligned} 1 \times 36 = 36 \\ 2 \times 18 = 36 \\ 3 \times 12 = 36 \\ 4 \times 9 = 36 \\ 6 \times 6 = 36 \end{aligned}

Now we can reverse the order of the factors for all of the ones listed above, because they are ordered pairs except 6 × 6 6 \times 6 since it is the same back if we reverse the order.

4 × 2 + 1 = 9 4 \times 2 + 1 = 9

(+1) for beating me in writing the solution ;-) ... Here's a b o n u s bonus -
How many pair of unordered M , N {\color{#D61F06}{M},\color{#3D99F6}{N}} satisfy the equation M N = 36 \color{#D61F06}{M}\color{#3D99F6}{N}=36 such that M , N Z {M,N}\in \mathbb{\textbf{Z}} .

Rishabh Jain - 5 years, 3 months ago

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There are 5 positive unordered pairs of M N MN and 5 negative, adding up to 10 pairs.

Arulx Z - 5 years, 3 months ago

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Exactly!! ..................

Rishabh Jain - 5 years, 3 months ago
Hanif Adzkiya
Mar 2, 2016

MN = 36 Prime factor of 36 = 2^2 * 3^2 Then, we can find that 36 has 9 factor, ... (2+1) (2+1)=3 3=9

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