P and Q are so close!

Algebra Level pending

P Q − Q P = P + Q P Q \frac{P}{Q} - \frac{Q}{P} = \frac{P+Q}{PQ}

Let P P and Q Q be in the interval [ 1 , 9 ] [1,9 ] . Find the total number of pairs of integers ( P , Q ) (P, Q) satisfying the equation above.

8 6 2 0 4

Fidel Simanjuntak by Fidel Simanjuntak , 19, Indonesia

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1 solution

P Q − Q P = P 2 − Q 2 P Q = P + Q P Q \frac{P}{Q} - \frac{Q}{P} = \frac{P^2-Q^2}{PQ} = \frac{P+Q}{PQ}

( P − Q ) ( P + Q ) P Q = P + Q P Q \frac{(P-Q)(P+Q)}{PQ} = \frac{P+Q}{PQ}

Simplify, we have

P − Q = 1 ⇒ P-Q = 1 \Rightarrow we get P > Q P > Q

Now, since 1 ≤ ( P , Q ) ≤ 9 1 \leq (P,Q) \leq 9 , we have

( P , Q ) = ( 2 , 1 ) , ( 3 , 2 ) , ( 4 , 3 ) , ( 5 , 4 ) , ( 6 , 5 ) , ( 7 , 6 ) , ( 8 , 7 ) , ( 9 , 8 ) (P,Q) = (2,1) , (3,2) , (4,3) , (5,4) , (6,5) , (7,6) , (8,7) , (9,8)

There are 8 \boxed{8} pairs.

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