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How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?
Correct answer is '4'. Can you explain this answer?
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How many pairs (m, n) of positive integers satisfy the equation m2 + ...
N2- m2 = 105
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(n-m) (n+m) = 1 *105, 3*35, 5*21, 7*15, 15*7, 21 *5, 35*3,105*1.
n – m=1, n+m=105==> n=53, m=52
n – m=3, n+m=35 ==> n=19, m=16
n – m=5, n+m=21 ==>n=13, m=8
n – m=7, n+m=15 ==> n=11, m=4
n – m=15, n+m=7 ==> n=11, m=-4
n – m=21, n+m=5 ==> n=13, m=-8
n – m=35, n+m=3 ==> n=19, m=-16
n – m=105, n+m=1 ==> n=53, m=-52
Since only positive integer values of m and n are required. There are 4 possible solutions.
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How many pairs (m, n) of positive integers satisfy the equation m2 + ...
Solution:
Given equation: m^2 - n^2 = 105
We can factorize the left-hand side of the equation as:
(m + n)(m - n) = 105
Since m and n are positive integers, we know that m + n > m - n. Therefore, we can write:
m + n > m - n ≥ 1
Thus, we have the following possibilities for the values of (m + n) and (m - n):
m + n = 105, m - n = 1
m + n = 35, m - n = 3
m + n = 21, m - n = 5
m + n = 15, m - n = 7
Solving each of these systems of equations, we get:
m = 53, n = 52
m = 19, n = 16
m = 13, n = 12
m = 11, n = 8
Therefore, there are 4 pairs of positive integers (m, n) that satisfy the given equation:
(53, 52), (19, 16), (13, 12), (11, 8)
Hence, the correct answer is 4.
Given equation: m^2 - n^2 = 105
We can factorize the left-hand side of the equation as:
(m + n)(m - n) = 105
Since m and n are positive integers, we know that m + n > m - n. Therefore, we can write:
m + n > m - n ≥ 1
Thus, we have the following possibilities for the values of (m + n) and (m - n):
m + n = 105, m - n = 1
m + n = 35, m - n = 3
m + n = 21, m - n = 5
m + n = 15, m - n = 7
Solving each of these systems of equations, we get:
m = 53, n = 52
m = 19, n = 16
m = 13, n = 12
m = 11, n = 8
Therefore, there are 4 pairs of positive integers (m, n) that satisfy the given equation:
(53, 52), (19, 16), (13, 12), (11, 8)
Hence, the correct answer is 4.
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How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer?
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How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer?.
How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer? for CAT 2024 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many pairs (m, n) of positive integers satisfy the equation m2 + 105 = n2?Correct answer is '4'. Can you explain this answer?.
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