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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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