Final answer:
To find the integers p and q satisfying 1/p - 1/q = 1/7, we first create a Diophantine equation and find that p = 8 and q = 56 are the valid positive integer solutions.
Step-by-step explanation:
The question asks to find two positive integers p and q that satisfy the equation 1/p - 1/q = 1/7.
To solve this, we can first find a common denominator for the fractions on the left side of the equation.
This common denominator would be pq. The equation then becomes:
(q - p) / pq = 1/7
Now, by cross-multiplying, we get:
q - p = pq / 7
This can be rearranged to form a Diophantine equation:
pq - 7q + 7p = 0
Which can be factored:
(p + 7)(q - 7) = 49
Now, since 49 is 7 squared and p and q are positive integers, we know that the pairs of factors of 49 are (1, 49) and (7, 7). However, to satisfy the equation, p + 7 and q - 7 must be these pairs. Thus:
p + 7 = 7 and q - 7 = 49
or
p + 7 = 1 and q - 7 = 49
The first case does not give a valid solution for a positive integer p, but the second case does:
p = 1 - 7 = -6 (not a valid positive integer)
q = 49 + 7 = 56
Since p must be positive and not -6, we discard the first case and only the second pair is valid which gives us:
p = 8 and q = 56. These values satisfy the original equation.