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How many ordered pairs of positive integers (m,n) satisfy ... 02 Apr 2019, 16:54
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Question Stats:

36% (02:12) correct 64% (02:09) wrong based on 263 sessions

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How many ordered pairs of positive integers \((m, n)\) satisfy the equation below?

\(|\sqrt{17} - m|\) \(+ \sqrt{17} + n = 20\)

A. 0
B. 10
C. 15
D. 19
E. 20
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C
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How many ordered pairs of positive integers (m,n) satisfy ... 02 Apr 2019, 22:16
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For the equation to work if m and n are integers, the √17 terms will need to disappear. That won't happen if √17 - m > 0, because then the absolute value will do nothing, and we'll have 2√17 on the left side of the equation. So it must be true that √17 - m < 0, and m > √17. In that case, |√17 - m| = -(√17 - m) = m - √17, and our equation becomes:

m - √17 + √17 + n = 20
m + n = 20

Since we know m > √17 and m is an integer, m must be 5 or greater, and since n is positive, m can take any integer value between 5 and 19 inclusive, so we have fifteen integer solutions.
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How many ordered pairs of positive integers (m,n) satisfy ... 03 Apr 2019, 05:00
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IanStewart
For the equation to work if m and n are integers, the √17 terms will need to disappear. That won't happen if √17 - m > 0, because then the absolute value will do nothing, and we'll have 2√17 on the left side of the equation. So it must be true that √17 - m < 0, and m > √17. In that case, |√17 - m| = -(√17 - m) = m - √17, and our equation becomes:

m - √17 + √17 + n = 20
m + n = 20

Since we know m > √17 and m is an integer, m must be 5 or greater, and since n is positive, m can take any integer value between 5 and 19 inclusive, so we have fifteen integer solutions.
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Hi,
Why should √17 - m > 0 or m - √17 < 0 ??
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How many ordered pairs of positive integers (m,n) satisfy ... 03 Apr 2019, 05:42
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IanStewart
For the equation to work if m and n are integers, the √17 terms will need to disappear. That won't happen if √17 - m > 0, because then the absolute value will do nothing, and we'll have 2√17 on the left side of the equation. So it must be true that √17 - m < 0, and m > √17. In that case, |√17 - m| = -(√17 - m) = m - √17, and our equation becomes:

m - √17 + √17 + n = 20
m + n = 20

Since we know m > √17 and m is an integer, m must be 5 or greater, and since n is positive, m can take any integer value between 5 and 19 inclusive, so we have fifteen integer solutions.

Hi,
Why should √17 - m > 0 or m - √17 < 0 ??
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Because \(|\sqrt{17} - m|\) is a modulus. You don't know whether is positive or not. So, you must consider both scenarios to derive the correct solution.
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How many ordered pairs of positive integers (m,n) satisfy ... 10 Jun 2019, 08:40
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IanStewart
For the equation to work if m and n are integers, the √17 terms will need to disappear. That won't happen if √17 - m > 0, because then the absolute value will do nothing, and we'll have 2√17 on the left side of the equation. So it must be true that √17 - m < 0, and m > √17. In that case, |√17 - m| = -(√17 - m) = m - √17, and our equation becomes:

m - √17 + √17 + n = 20
m + n = 20

Since we know m > √17 and m is an integer, m must be 5 or greater, and since n is positive, m can take any integer value between 5 and 19 inclusive, so we have fifteen integer solutions.
Show more

Why m can take value between 5 and 19?
Why can't we take \(m = 20\)?
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How many ordered pairs of positive integers (m,n) satisfy ... 10 Jun 2019, 08:44
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Abhishek292kumar

Why m can take value between 5 and 19?
Why can't we take \(m = 20\)?
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If m + n = 20, m cannot be 20, because then n would need to be zero, and we are told that n is a positive integer.
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How many ordered pairs of positive integers (m,n) satisfy ... 28 Mar 2020, 07:59
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lucajava
How many ordered pairs of positive integers \((m, n)\) satisfy the equation below?

\(|\sqrt{17} - m|\) \(+ \sqrt{17} + n = 20\)

A. 0
B. 10
C. 15
D. 19
E. 20
Show more

Asked: How many ordered pairs of positive integers \((m, n)\) satisfy the equation below?

\(|\sqrt{17} - m|\) \(+ \sqrt{17} + n = 20\)

Since m & n are positive integers
\(|\sqrt{17} - m| = m -\sqrt{17} \)
\(m>=\sqrt{17}>4\)

m + n = 20

(m,n)={(5,15),(6,14),,,,,(19,1)}

Number of ordered pairs = 19-5 + 1 = 15

IMO C
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