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How many pairs of positive integers (m,n) satisfy m+ n < 22?

Feb 16, 2020

#1
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The first step to solving this problem is to figure out sample values of m. $$m$$ can be 1, 2, 3, or 4. 5 would be too large as 5 squared is 25 and 22-25 is a negative number, it says ONLY positive integers. 0 isn't a positive number. So there are more than 4 pairs of positive integers (m, n) that satisfy the equation $$m^2 +n <22$$. Then figure out n for each m.

Feb 16, 2020
edited by Hypotenuisance  Feb 16, 2020
edited by Hypotenuisance  Feb 16, 2020
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Thank you Hypotenuisance! That was the correct answer 😊

Guest Feb 16, 2020
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@Guest Sure, no problem!

Hypotenuisance  Feb 16, 2020
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If you want the answer, I believe it is 54 pairs.

Feb 16, 2020
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How many pairs of positive integers (m,n) satisfy $$m^2 + n < 22$$ ?

Hello Guest!

$$\{m,n\} \subset\mathbb N\\ \mathbb L=m^2+n\\ \mathbb L\in \mathbb \{\mathbb R<22\}$$

$$n=1\to m\in \{1,2,3,4 \}\\ n=2\to m\in \{1,2,3,4 \}\\ n=3\to m\in \{1,2,3,4 \}\\ n=4\to m\in \{1,2,3,4 \}\\ n=5\to m\in \{1,2,3,4 \}$$

$$n=6\to m\in \{1,2,3 \}\\ n=7\to m\in \{1,2,3 \}\\ n=8\to m\in \{1,2,3 \}\\ n=9\to m\in \{1,2,3 \}\\ n=10\to m\in \{1,2,3 \}\\ n=11\to m\in \{1,2,3\}$$

$$n=12\to m\in \{1,2,3 \}\\ n=13\to m\in \{1,2 \}\\ n=14\to m\in \{1,2 \}\\ n=15\to m\in \{1,2 \}\\ n=16\to m\in \{1,2 \}$$

$$n=17\to m\in \{1,2 \}\\ n=18\to m=1\\ n=19\to m=1 \\ n=20\to m=1\\$$

53 pairs of positive integers (m,n) satisfy m2 + n < 22

!

Feb 16, 2020
edited by asinus  Feb 16, 2020
edited by asinus  Feb 17, 2020