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DIRECTIONS for the question: Solve the following question and mark the best possible option.
The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.
- a)1
- b)2
- c)3
- d)4
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
DIRECTIONS for the question:Solve the following question and mark the ...
Given 4x + 5y = 134
► 5mx + 80 = 134 - 4x ⇒ (5m + 4) = 54
Since m ∈ N, x ∈ N.
► Now consider the factors of 54 i.e., 1, 2, 3, 6, 9,18, 27 and 54.
► Out of these only 9 and 54 can be expressed as 5m + 4
► i.e., m = 1 and m = 10 are the two possible values.
Since m ∈ N, x ∈ N.
► Now consider the factors of 54 i.e., 1, 2, 3, 6, 9,18, 27 and 54.
► Out of these only 9 and 54 can be expressed as 5m + 4
► i.e., m = 1 and m = 10 are the two possible values.
Most Upvoted Answer
DIRECTIONS for the question:Solve the following question and mark the ...
Given:
- The lines 4x - 5y = 134 and y = mx - 16 intersect at points whose coordinates are integers.
To find:
- The number of positive integer values that m can assume.
Solution:
To find the number of positive integer values that m can assume, we need to find the intersection points of the two given lines and check if the coordinates of the intersection points are integers.
Step 1: Find the intersection point
To find the intersection point, we will equate the two equations:
4x - 5y = 134 ...(1)
y = mx - 16 ...(2)
Substituting equation (2) into equation (1), we get:
4x - 5(mx - 16) = 134
4x - 5mx + 80 = 134
4x - 5mx = 54
x(4 - 5m) = 54
x = 54 / (4 - 5m)
Step 2: Check if the coordinates are integers
For the coordinates to be integers, both x and y must be integers.
From equation (2), we have y = mx - 16. Since x is an integer, y will also be an integer if and only if mx - 16 is an integer.
Substituting x = 54 / (4 - 5m) into the equation, we get:
y = (54 / (4 - 5m)) * m - 16
y = (54m / (4 - 5m)) - 16
For y to be an integer, the numerator (54m) must be divisible by the denominator (4 - 5m).
Step 3: Find the possible values of m
To find the possible values of m, we need to find the values of m that satisfy the condition:
54m is divisible by (4 - 5m)
We can check the possible values of m by substituting different positive integers for m and checking if the condition is satisfied.
Checking for m = 1:
54 * 1 = 54
4 - 5 * 1 = -1
54 is divisible by -1: Not satisfied
Checking for m = 2:
54 * 2 = 108
4 - 5 * 2 = -6
108 is divisible by -6: Not satisfied
Checking for m = 3:
54 * 3 = 162
4 - 5 * 3 = -11
162 is divisible by -11: Not satisfied
Checking for m = 4:
54 * 4 = 216
4 - 5 * 4 = -16
216 is divisible by -16: Not satisfied
Checking for m = 5:
54 * 5 = 270
4 - 5 * 5 = -21
270 is divisible by -21: Not satisfied
Checking for m = 6:
54 * 6 = 324
4 - 5 * 6 = -26
324 is divisible by -26: Not satisfied
Checking for m = 7:
54 * 7 =
- The lines 4x - 5y = 134 and y = mx - 16 intersect at points whose coordinates are integers.
To find:
- The number of positive integer values that m can assume.
Solution:
To find the number of positive integer values that m can assume, we need to find the intersection points of the two given lines and check if the coordinates of the intersection points are integers.
Step 1: Find the intersection point
To find the intersection point, we will equate the two equations:
4x - 5y = 134 ...(1)
y = mx - 16 ...(2)
Substituting equation (2) into equation (1), we get:
4x - 5(mx - 16) = 134
4x - 5mx + 80 = 134
4x - 5mx = 54
x(4 - 5m) = 54
x = 54 / (4 - 5m)
Step 2: Check if the coordinates are integers
For the coordinates to be integers, both x and y must be integers.
From equation (2), we have y = mx - 16. Since x is an integer, y will also be an integer if and only if mx - 16 is an integer.
Substituting x = 54 / (4 - 5m) into the equation, we get:
y = (54 / (4 - 5m)) * m - 16
y = (54m / (4 - 5m)) - 16
For y to be an integer, the numerator (54m) must be divisible by the denominator (4 - 5m).
Step 3: Find the possible values of m
To find the possible values of m, we need to find the values of m that satisfy the condition:
54m is divisible by (4 - 5m)
We can check the possible values of m by substituting different positive integers for m and checking if the condition is satisfied.
Checking for m = 1:
54 * 1 = 54
4 - 5 * 1 = -1
54 is divisible by -1: Not satisfied
Checking for m = 2:
54 * 2 = 108
4 - 5 * 2 = -6
108 is divisible by -6: Not satisfied
Checking for m = 3:
54 * 3 = 162
4 - 5 * 3 = -11
162 is divisible by -11: Not satisfied
Checking for m = 4:
54 * 4 = 216
4 - 5 * 4 = -16
216 is divisible by -16: Not satisfied
Checking for m = 5:
54 * 5 = 270
4 - 5 * 5 = -21
270 is divisible by -21: Not satisfied
Checking for m = 6:
54 * 6 = 324
4 - 5 * 6 = -26
324 is divisible by -26: Not satisfied
Checking for m = 7:
54 * 7 =
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DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?
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DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. 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 DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?.
DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. 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 DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for CAT 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for DIRECTIONS for the question:Solve the following question and mark the best possible option.The lines 4x + 5y = 134 and y = mx + 16 intersect at points whose coordinates are integers. Find the number of positive integer values that m can assume.a)1b)2c)3d)4Correct answer is option 'B'. Can you explain this answer?.
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