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Try this beautiful problem from Algebra: Order Pair

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

- \(31\)
- \(78\)
- \(43\)

Algebra

Order Pair

Multiplication

But try the problem first...

Answer: \(78\)

Source

Suggested Reading

AMC-10A (2010) Problem 21

Pre College Mathematics

First hint

Given that $\frac{M}{6}=\frac{6}{N}$ \(\Rightarrow MN=36\).Next we have to find out the the Possibilities to getting \(a \times b=36\)

can you finish the problem........

Second Hint

Now the possibilities are ....

$1 \times 36=36$

$2 \times 18=36$

$3 \times 12=36$

$4 \times 9=36$

$6 \times 6=36$

We can reverse the order of the factors for all of the ones listed above, because they are ordered pairs except for 6*6 since it is the same back if you reverse the order.

can you finish the problem........

Final Step

Therefore the total Possible order pairs that satisfy the equation $\frac{M}{6}=\frac{6}{N}$=\(9\)

- https://www.cheenta.com/problem-based-on-triangle-prmo-2016-problem-10/
- https://www.youtube.com/watch?v=VLyrlx2DWdA&t=17s

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