Find the number of ordered quadruples of positive integers (a,b,c,d ) such that ab +cd =10.
Why did the positive integers go to couples therapy? Because they couldn't figure out how to make their relationship add up to 10!
To find the number of ordered quadruples of positive integers (a,b,c,d) such that ab + cd = 10, we can consider the possible values of (a, b) and (c, d).
Firstly, let's list out all the possible pairs of positive integers (a, b) that multiply to give 10:
(1, 10), (2, 5), (5, 2), (10, 1)
For each of these pairs, we need to find the possible pairs (c, d) that also multiply to give 10 - ab.
For (a, b) = (1, 10):
(1, 10) can be paired with (1, 10), (2, 5), (5, 2), or (10, 1)
For (a, b) = (2, 5):
(2, 5) can be paired with (1, 10) or (2, 5)
For (a, b) = (5, 2):
(5, 2) can be paired with (1, 10) or (2, 5)
For (a, b) = (10, 1):
(10, 1) can only be paired with (1, 10)
Therefore, the ordered quadruples of positive integers (a, b, c, d) that satisfy ab + cd = 10 are as follows:
(1, 10, 1, 10)
(1, 10, 2, 5)
(1, 10, 5, 2)
(1, 10, 10, 1)
(2, 5, 1, 10)
(2, 5, 2, 5)
(5, 2, 1, 10)
(5, 2, 2, 5)
(10, 1, 1, 10)
Hence, there are a total of 9 ordered quadruples of positive integers (a, b, c, d) that satisfy ab + cd = 10.
To find the number of ordered quadruples of positive integers (a, b, c, d) such that ab + cd = 10, we can consider different cases and use some basic combinatorics.
Case 1: a=1
If a = 1, then we have b + cd = 10. To find the number of solutions for this equation, we can iterate through all possible values of b and find the corresponding value of c and d. Since a and b are positive integers, the possible values of b are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. For each value of b, we can calculate the corresponding value of c by subtracting b from 10 (since b + c = 10), and then find the value of d as well. Therefore, there are 10 possible quadruples for this case.
Case 2: a=2
If a = 2, then we have 2b + cd = 10. Similarly, we can iterate through possible values of b and find corresponding values of c and d. The possible values for b are 1, 2, 3, 4, and 5. For each value of b, we can calculate the corresponding value of cd by subtracting 2b from 10, and then find the values of c and d. In this case, we get the following quadruples: (2, 4, 1, 4), (2, 3, 2, 2), (2, 2, 3, 1), (2, 1, 5, 0), and (2, 0, 10, 0). Therefore, there are 5 possible quadruples for this case.
Case 3: a=3
If a = 3, then we have 3b + cd = 10. We can follow a similar process as before and iterate through possible values of b to find corresponding values of c and d. The possible values for b are 1 and 2. For each value of b, we can calculate the corresponding value of cd by subtracting 3b from 10, and then find the values of c and d. In this case, we get the following quadruples: (3, 2, 1, 1) and (3, 1, 2, 0). Therefore, there are 2 possible quadruples for this case.
Case 4: a=4
If a = 4, then we have 4b + cd = 10. Following the same process, we find that there is only one possible quadruple for this case: (4, 1, 1, 6).
Case 5: a=5
If a = 5, then we have 5b + cd = 10. Similarly, we can iterate through possible values of b and find corresponding values of c and d. The possible value for b is 1, and for this value, we can calculate the corresponding value of cd by subtracting 5b from 10, and then find the values of c and d. In this case, we get the following quadruple: (5, 1, 1, 1). Therefore, there is only one possible quadruple for this case.
By considering all the cases, we find a total of 10 + 5 + 2 + 1 + 1 = 19 possible quadruples that satisfy the given equation.
so you have
1+9=10
2+8=10
...
9+1=10
consider 4+6 = 10
to get 4 we have 1x4 , 2x2 , 4x1
to get 6 we have 1x6, 2x3, 3x2, 6x1
so one possible quadruple is (2,2,3,2)
start grinding them out