Number of solutions

Dec 2015
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166
Earth
Find the number of pairs \(\displaystyle (n_1,n_2)\) that satisfies the equation below:
\(\displaystyle n_1 + n_2 = x\; \; \) where \(\displaystyle n_1 ,n_2 ,x \in \mathbb{N}\)
 
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topsquark

Math Team
May 2013
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1,048
The Astral plane
Find the number of pairs \(\displaystyle (n_1,n_2)\) that satisfies the equation below:
\(\displaystyle n_1 + n_2 = x\; \; \) where \(\displaystyle n_1 ,n_2 ,x \in \mathbb{N}\)
Let's rephrase the problem statement a bit.
\(\displaystyle n_1 + n_2 = x\)

\(\displaystyle n_2 = x - n_1\)

So
\(\displaystyle n_1 = 1\), \(\displaystyle n_2 = x - 1\).

\(\displaystyle n_1 = 2\), \(\displaystyle n_2 = x - 2\).

And now it's a counting problem. How many values can \(\displaystyle n_1\) take on?

-Dan
 
Dec 2015
1,069
166
Earth
It is seen that \(\displaystyle x\geq 2\)
Number of pairs is dependent on \(\displaystyle x\)
Example , \(\displaystyle x=10^{10}\)
 

topsquark

Math Team
May 2013
2,508
1,048
The Astral plane
It is seen that \(\displaystyle x\geq 2\)
Number of pairs is dependent on \(\displaystyle x\)
Example , \(\displaystyle x=10^{10}\)
Yes. Though I'm not going to try to list all that with your x value!

-Dan
 
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Dec 2015
1,069
166
Earth
What i got is \(\displaystyle f(x)=x-1\)
Where f(x) is the number of pairs