How many ordered pairs of positive integers (a,b) are there such that (4a-1)/b and (4b-1)/a are both integers?
clearly (1,1) will work
I then made up and ran a silly little computer program where I let a go from 1 to 1000 and b = 1 to 1000 and tested each case
(so my program tested 1 million cases)
it found the following ordered pairs
1 3 3 1 5 19 7 9 9 7 19 5
testing (19,5)
if a = 19 and b=5
(4a-1)/b = 75/5 = 15 which is an integer
(4b-1)/a = 19/19 = 1 , which is an integer
notice they come in symmetric pairs
so there 7 of these.
I just changed my a values from 1 to 10,000
and my b value from 1 to 10,000
and it found no others.
should have put them in ordered pairs:
(1,1) (1,3) (3,1) (5,19) , (7,9), (9,7), (19,5)
To find the number of ordered pairs of positive integers (a, b) that satisfy the given condition, we need to analyze the problem step by step.
First, let's check the divisibility of (4a - 1) by b. If (4a - 1) is divisible by b, then (4a - 1)/b is an integer. Similarly, if (4b - 1) is divisible by a, then (4b - 1)/a is an integer.
To simplify the problem, we can consider the cases where (4a - 1) and (4b - 1) are divisible by smaller primes. Let's start by checking divisibility by 2 and 3.
Case 1: Divisibility by 2
If (4a - 1) is divisible by 2, then (4a - 1) must be odd. This means that 'a' must be odd. So, a can take values of 1, 3, 5, 7, ...
If (4b - 1) is divisible by 2, then (4b - 1) must be odd. This means that 'b' must also be odd. So, b can take values of 1, 3, 5, 7, ...
Since both 'a' and 'b' can take the same set of odd values, let's consider the number of possible common odd values of 'a' and 'b'.
Considering the range of values for 'a' and 'b' as positive integers, let's try to find the number of ordered pairs (a, b) with common odd values up to a certain limit.
If we choose a limit of 'N' for both 'a' and 'b', then the number of odd values up to 'N' is given by (N + 1) / 2 (due to the odd numbers being equally spaced).
So, for 'a' and 'b' both ranging up to 'N', the number of ordered pairs (a, b) with common odd values is ((N + 1) / 2) * ((N + 1) / 2) = ((N + 1) / 2)^2.
Case 2: Divisibility by 3
If (4a - 1) is divisible by 3, then (4a - 1) must be congruent to 0 modulo 3, which implies (4a) must be congruent to 1 modulo 3. In other words, 'a' must be congruent to 3 modulo 3. So, 'a' can take values of 3, 6, 9, 12, ...
If (4b - 1) is divisible by 3, then (4b - 1) must be congruent to 0 modulo 3, which implies (4b) must be congruent to 1 modulo 3. In other words, 'b' must be congruent to 3 modulo 3. So, 'b' can take values of 3, 6, 9, 12, ...
Since both 'a' and 'b' can take the same set of values congruent to 3 modulo 3, we can again consider the number of ordered pairs (a, b) with common values.
Similarly, for each modulus m, we can find the number of ordered pairs (a, b) with common values.
In conclusion, by considering the divisibility of (4a - 1) and (4b - 1) by smaller primes and finding the number of ordered pairs (a, b) with common values according to each modulus, we can calculate the total number of ordered pairs (a, b) that satisfy the given condition.