what-is-the-number-of-ordered-pairs-of-positif-integers-x-y-that-satisfy-x-2-y-2-xy-37-

Question Number 100178 by bobhans last updated on 25/Jun/20

what is the number of ordered pairs of positif

integers (x, y) that satisfy x^{2} + y^{2} - xy = 37

Answered by 1549442205 last updated on 25/Jun/20

x^{2} - xy + y^{2} - 37 = 0 . We look at it like a

quadratic equation respect with x . Then

Δ = y^{2} - {4 y}^{2} + 148 = - {3 y}^{2} + 148 . The above

eqs . always has roots, so Δ = - {3 y}^{2} + 148 ⩾ 0

\Rightarrow y^{2} ⩽ \frac{148}{3} . Since y \in N^{*}, y \in {1, 2, \dots, 7} .

i) If y \in {1, 2, 5, 6} then Δ \in {145, 136, 73, 40}

{aren}^{'} t perfect numbers, so are rejected

ii) for y = 3 \Rightarrow Δ = 121 = 11^{2} \Rightarrow x = \frac{3 + 11}{2} = 7 (we reject negative root)

we get the root (x, y) = (7; 3)

iii) for y = 4 \Rightarrow Δ = 100 = 10^{2} \Rightarrow x = \frac{4 + 10}{2} = 7

we get the root (x; y) = (7; 4)

iv) for y = 7 \Rightarrow Δ = 1 \Rightarrow x = \frac{7 \pm 1}{2} \Rightarrow x \in {4; 3}

we get two roots (x; y) \in {(4; 7); (3; 7)}

Thus, the given equation has four roots

(x; y) \in {(7; 3); (7; 4); (3; 7); (4; 7)}

Commented by bemath last updated on 25/Jun/20

how with (4, - 3), (- 4, 3), (3, - 4), (- 3, 4) ?

Commented by 1549442205 last updated on 25/Jun/20

your problem require to find the pairs of

positive integers

Answered by Rasheed.Sindhi last updated on 25/Jun/20

\underline{A Novel Method}

x^{2} + y^{2} - xy = 37

{(x - y)}^{2} = 37 - xy \dots \dots \dots \dots A

{(x + y)}^{2} = 37 + 3 xy \dots \dots \dots \dots B

x, y \in Z^{+} \land 37 - xy is perfect square

37 - xy = 36, 25, 16, 9, 4, 1 (possible values)

xy = 1, 12, 21, 28, 33, 36

From above values of xy acceptable

values are those for which 37 + 3 xy

is also perfect square are 21 & 28

When xy = 21

A \Rightarrow x - y = 4 (assume x ⩾ y)

B \Rightarrow x + y = 10

x = 7, y = 3

(or x = 3, y = 7 due to symmetry)

When xy = 28

A \Rightarrow x - y = 3

B \Rightarrow x + y = 11

x = 7, y = 4

(or x = 4, y = 7 due to symmetry)

(7, 3), (3, 7), (7, 4), (4, 7)

Commented by mr W last updated on 25/Jun/20

n i c e s o l u t i o n s i r!

Commented by bobhans last updated on 26/Jun/20

thank you sir

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