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Two positive integers p and q satisfy the equation 1/p-1/q=1/7. Find p and q

User Yamuk
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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.

User Keith OYS
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