To find the smallest positive integer that satisfies these conditions, we can use the Chinese Remainder Theorem.
The Chinese Remainder Theorem states that if we have a set of congruences:
x ā” aā (mod nā)
x ā” aā (mod nā)
...
x ā” aā (mod nā)
where nā, nā, ..., nā are pairwise coprime, then there exists a unique solution modulo (nā * nā * ... * nā).
In this case, we have the following congruences:
x ā” 5 (mod 7)
x ā” 6 (mod 11)
x ā” 4 (mod 13)
The moduli, 7, 11, and 13, are pairwise coprime, so we can apply the Chinese Remainder Theorem.
To find the solution, we can use the Extended Euclidean Algorithm. Let's solve for the first pair of congruences:
x ā” 5 (mod 7)
x ā” 6 (mod 11)
By using the Extended Euclidean Algorithm, we can find coefficients s and t such that:
7s + 11t = 1
We can solve this equation to find s and t:
7s + 11t = 1
Solving this equation, we find that s = 3 and t = -2.
Now, we can express the solution as:
x = (5 * 11 * -2 + 6 * 7 * 3) mod (7 * 11)
x = (-110 + 126) mod 77
x = 16 mod 77
So, the solution for the first pair of congruences is x ā” 16 (mod 77).
Now, we can solve the second pair of congruences:
x ā” 16 (mod 77)
x ā” 4 (mod 13)
Using the Extended Euclidean Algorithm, we can find coefficients s and t such that:
77s + 13t = 1
Solving this equation, we find that s = 5 and t = -29.
Now, we can express the solution as:
x = (16 * 13 * -29 + 4 * 77 * 5) mod (77 * 13)
x = (-6032 + 1540) mod 1001
x = -4492 mod 1001
x = 509 mod 1001
So, the smallest positive integer that satisfies all three congruences is x = 509.