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How many pairs (m, n) of positive integers satisfy the equation m2 +105 = n2?
[2019]
- a)5
- b)4
- c)3
- d)2
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
How many pairs (m, n) of positive integers satisfy the equation m2 +10...
m2 + 105 = n2 ⇒ n2 – m2 = 105
⇒ (n – m) (n + m) = 105
Since m an d n are positive integers (n – m) < (n + m), then by splitting 105 in two factors, we get
⇒ (n – m) (n + m) = 1 × 105
For (n – m) = 1 and (n + m) = 105, (m, n) = (52, 53)
⇒ (n – m) (n + m) = 3 × 35
For (n – m) = 3 and (n + m) = 35 (m, n) = (16, 19)
⇒ (n – m) (n + m) = 5 × 21
For (n – m) = 5 and (n + m) = 21 (m, n) = (8, 13)
⇒ (n – m) (n + m) = 7 × 15
For (n – m) = 7 and (n + m) = 15, (m, n) = (4, 11)
Hence, there are four required pairs.
Shortcut approach :
Number of pairs =
105 = 3 × 5 × 7
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
Number of factors of 105 = 8
Hence, required number of pairs = 8 / 2 = 4
⇒ (n – m) (n + m) = 105
Since m an d n are positive integers (n – m) < (n + m), then by splitting 105 in two factors, we get
⇒ (n – m) (n + m) = 1 × 105
For (n – m) = 1 and (n + m) = 105, (m, n) = (52, 53)
⇒ (n – m) (n + m) = 3 × 35
For (n – m) = 3 and (n + m) = 35 (m, n) = (16, 19)
⇒ (n – m) (n + m) = 5 × 21
For (n – m) = 5 and (n + m) = 21 (m, n) = (8, 13)
⇒ (n – m) (n + m) = 7 × 15
For (n – m) = 7 and (n + m) = 15, (m, n) = (4, 11)
Hence, there are four required pairs.
Shortcut approach :
Number of pairs =
105 = 3 × 5 × 7
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
Number of factors of 105 = 8
Hence, required number of pairs = 8 / 2 = 4
Most Upvoted Answer
How many pairs (m, n) of positive integers satisfy the equation m2 +10...
m2 + 105 = n2 ⇒ n2 – m2 = 105
⇒ (n – m) (n + m) = 105
Since m an d n are positive integers (n – m) < (n + m), then by splitting 105 in two factors, we get
⇒ (n – m) (n + m) = 1 × 105
For (n – m) = 1 and (n + m) = 105, (m, n) = (52, 53)
⇒ (n – m) (n + m) = 3 × 35
For (n – m) = 3 and (n + m) = 35 (m, n) = (16, 19)
⇒ (n – m) (n + m) = 5 × 21
For (n – m) = 5 and (n + m) = 21 (m, n) = (8, 13)
⇒ (n – m) (n + m) = 7 × 15
For (n – m) = 7 and (n + m) = 15, (m, n) = (4, 11)
Hence, there are four required pairs.
Shortcut approach :
Number of pairs =
105 = 3 × 5 × 7
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
Number of factors of 105 = 8
Hence, required number of pairs = 8 / 2 = 4
⇒ (n – m) (n + m) = 105
Since m an d n are positive integers (n – m) < (n + m), then by splitting 105 in two factors, we get
⇒ (n – m) (n + m) = 1 × 105
For (n – m) = 1 and (n + m) = 105, (m, n) = (52, 53)
⇒ (n – m) (n + m) = 3 × 35
For (n – m) = 3 and (n + m) = 35 (m, n) = (16, 19)
⇒ (n – m) (n + m) = 5 × 21
For (n – m) = 5 and (n + m) = 21 (m, n) = (8, 13)
⇒ (n – m) (n + m) = 7 × 15
For (n – m) = 7 and (n + m) = 15, (m, n) = (4, 11)
Hence, there are four required pairs.
Shortcut approach :
Number of pairs =
105 = 3 × 5 × 7
Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and 105
Number of factors of 105 = 8
Hence, required number of pairs = 8 / 2 = 4
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Community Answer
How many pairs (m, n) of positive integers satisfy the equation m2 +10...
To solve the equation m^2 - 105 = n^2, we need to find the number of pairs (m, n) of positive integers that satisfy the equation.
1. Understanding the equation:
The equation can be rearranged as m^2 - n^2 = 105, which is a difference of squares. This can be further factored as (m + n)(m - n) = 105.
2. Considering the factors of 105:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. We can consider all possible combinations of these factors to find the pairs (m, n) that satisfy the equation.
3. Exploring the factors:
Let's consider each factor of 105 and solve the resulting equations to find the possible values of m and n.
a) m + n = 105, m - n = 1:
Adding these two equations gives us 2m = 106, which implies m = 53. Substituting this value into either equation gives n = 52.
So, one pair (m, n) is (53, 52).
b) m + n = 35, m - n = 3:
Adding these two equations gives us 2m = 38, which implies m = 19. Substituting this value into either equation gives n = 16.
So, another pair (m, n) is (19, 16).
c) m + n = 21, m - n = 5:
Adding these two equations gives us 2m = 26, which implies m = 13. Substituting this value into either equation gives n = 8.
So, another pair (m, n) is (13, 8).
d) m + n = 15, m - n = 7:
Adding these two equations gives us 2m = 22, which implies m = 11. Substituting this value into either equation gives n = 4.
So, another pair (m, n) is (11, 4).
4. Counting the pairs:
From the above calculations, we can see that there are four pairs (m, n) of positive integers that satisfy the equation m^2 - 105 = n^2.
Therefore, the correct answer is option 'B' (4).
1. Understanding the equation:
The equation can be rearranged as m^2 - n^2 = 105, which is a difference of squares. This can be further factored as (m + n)(m - n) = 105.
2. Considering the factors of 105:
The factors of 105 are 1, 3, 5, 7, 15, 21, 35, and 105. We can consider all possible combinations of these factors to find the pairs (m, n) that satisfy the equation.
3. Exploring the factors:
Let's consider each factor of 105 and solve the resulting equations to find the possible values of m and n.
a) m + n = 105, m - n = 1:
Adding these two equations gives us 2m = 106, which implies m = 53. Substituting this value into either equation gives n = 52.
So, one pair (m, n) is (53, 52).
b) m + n = 35, m - n = 3:
Adding these two equations gives us 2m = 38, which implies m = 19. Substituting this value into either equation gives n = 16.
So, another pair (m, n) is (19, 16).
c) m + n = 21, m - n = 5:
Adding these two equations gives us 2m = 26, which implies m = 13. Substituting this value into either equation gives n = 8.
So, another pair (m, n) is (13, 8).
d) m + n = 15, m - n = 7:
Adding these two equations gives us 2m = 22, which implies m = 11. Substituting this value into either equation gives n = 4.
So, another pair (m, n) is (11, 4).
4. Counting the pairs:
From the above calculations, we can see that there are four pairs (m, n) of positive integers that satisfy the equation m^2 - 105 = n^2.
Therefore, the correct answer is option 'B' (4).
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How many pairs (m, n) of positive integers satisfy the equation m2 +105 = n2?[2019]a)5b)4c)3d)2Correct answer is option 'B'. Can you explain this answer?
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