1. If b is positive integer less than 200, then how many integer pairs (a,b) satisfy the equation a/b= 2/9?
done!
patience!, you posted the same question only about 8 minutes before this one.
That was by mistake, I didn't know and typed in again.
To solve this problem, we need to find the number of integer pairs (a, b) that satisfy the equation a/b = 2/9, where b is a positive integer less than 200.
To begin, let's rewrite the equation by cross-multiplying:
9a = 2b
Next, we need to analyze the equation to determine the values of a and b that satisfy it. Since b is a positive integer less than 200, we can limit our analysis to positive integers between 1 and 199.
Let's rearrange the equation to isolate a:
a = (2b)/9
Given that a must be an integer, we need the numerator, 2b, to be divisible by 9. Hence, b must be divisible by 9.
Now, let's look at the range of possible values for b. We want b to be a positive integer less than 200, and also divisible by 9. The first positive integer that satisfies these conditions is 9. The largest positive integer divisible by 9 and less than 200 is 198.
Therefore, the possible values for b are 9, 18, 27, ..., 198. To find the number of integer pairs, we need to determine the count of these possible values for b.
To find the count, we can use the formula for an arithmetic sequence:
count = (last term - first term)/common difference + 1
The first term is 9, the last term is 198, and the common difference is 9. Plugging these values into the formula, we get:
count = (198 - 9)/9 + 1
count = 21
Therefore, there are 21 integer pairs (a, b) that satisfy the equation a/b = 2/9, where b is a positive integer less than 200.