35. If b is positive integers less than 400 and more than 100, then how many integer pairs (a,b) satisfy the equation a/b=2/9?
To find the number of integer pairs (a, b) that satisfy the equation a/b = 2/9, we need to analyze the given conditions.
First, we know that b is a positive integer less than 400 and more than 100. This means that b can take values from 101 to 399 inclusive.
Next, let's consider the equation a/b = 2/9. Since a and b are integers, we can rewrite this equation as a = (2/9) * b.
To find the valid integer pairs (a, b), we need to find values of b that result in an integer value of a.
Multiplying both sides of the equation by 9 to eliminate the fraction, we get 9a = 2b.
Now, we know that a = 2b/9. For a to be an integer, the numerator (2b) needs to be a multiple of 9.
Since 2b is even, it must be divisible by 2 as well. Therefore, if a is an integer, b must also be divisible by 2.
Now, let's consider the valid values of b within the given range (101 to 399) that satisfy the conditions:
101 is not divisible by 2, so it's not a valid value of b.
102 is divisible by 2, so it's a valid value of b. In this case, a = 2b/9 = 2(102)/9 = 68.
103 is not divisible by 2, so it's not a valid value of b.
104 is divisible by 2, so it's a valid value of b. In this case, a = 2b/9 = 2(104)/9 = 23.
...
Continuing this pattern, we can find the other valid values of b and corresponding values of a.
Since b varies from 101 to 399 by steps of 2, we can calculate the number of valid integer pairs by counting the number of valid b values.
The series of valid b values goes as follows: 102, 104, 106, 108, ..., 398.
To find the number of terms in this series, we can subtract the first term from the last term and divide the result by the step size (which is 2 in this case).
(398 - 102) / 2 = 296 / 2 = 148.
Therefore, there are 148 integer pairs (a, b) that satisfy the given equation a/b = 2/9, where b is a positive integer less than 400 and more than 100.