Order Pair | AMC-10B, 2012 | Problem 10

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Try this beautiful problem from Algebra: Order Pair

Order Pair - AMC-10B, 2012- Problem 10

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

$31$
$78$
$43$

Key Concepts

Algebra

Order Pair

Multiplication

Check the Answer

But try the problem first...

Answer: $78$

Source

Try with Hints

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 $\frac{M}{6}=\frac{6}{N}$=$9$