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How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?
- a)0
- b)2
- c)3
- d)4
Correct answer is option 'D'. Can you explain this answer?
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How many positive integer pairs (a, b) satisfy the equation ab = a + b...
Problem Analysis:
We are given the equation ab = a + b + 20 and we need to find the number of positive integer pairs (a, b) that satisfy this equation.
Solution:
To solve this problem, let's rewrite the given equation in a different form:
ab - a - b = 20
Now, let's add 1 on both sides of the equation to factorize the left-hand side:
ab - a - b + 1 = 20 + 1
(a - 1)(b - 1) = 21
Factorizing 21:
The factors of 21 are:
1 x 21 = 21
3 x 7 = 21
So, we have two cases to consider for the factorization of 21.
Case 1: (a - 1) = 1 and (b - 1) = 21
If (a - 1) = 1, then a = 2
If (b - 1) = 21, then b = 22
So, the first pair (a, b) = (2, 22) satisfies the equation.
Case 2: (a - 1) = 3 and (b - 1) = 7
If (a - 1) = 3, then a = 4
If (b - 1) = 7, then b = 8
So, the second pair (a, b) = (4, 8) satisfies the equation.
Therefore, there are two positive integer pairs (2, 22) and (4, 8) that satisfy the given equation.
Answer:
The correct answer is option D) 4, as there are two positive integer pairs that satisfy the equation.
We are given the equation ab = a + b + 20 and we need to find the number of positive integer pairs (a, b) that satisfy this equation.
Solution:
To solve this problem, let's rewrite the given equation in a different form:
ab - a - b = 20
Now, let's add 1 on both sides of the equation to factorize the left-hand side:
ab - a - b + 1 = 20 + 1
(a - 1)(b - 1) = 21
Factorizing 21:
The factors of 21 are:
1 x 21 = 21
3 x 7 = 21
So, we have two cases to consider for the factorization of 21.
Case 1: (a - 1) = 1 and (b - 1) = 21
If (a - 1) = 1, then a = 2
If (b - 1) = 21, then b = 22
So, the first pair (a, b) = (2, 22) satisfies the equation.
Case 2: (a - 1) = 3 and (b - 1) = 7
If (a - 1) = 3, then a = 4
If (b - 1) = 7, then b = 8
So, the second pair (a, b) = (4, 8) satisfies the equation.
Therefore, there are two positive integer pairs (2, 22) and (4, 8) that satisfy the given equation.
Answer:
The correct answer is option D) 4, as there are two positive integer pairs that satisfy the equation.
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How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer?
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How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer?.
How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for How many positive integer pairs (a, b) satisfy the equation ab = a + b + 20?a)0b)2c)3d)4Correct answer is option 'D'. Can you explain this answer?.
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