Let-x-denote-the-greatest-integer-x-Then-the-number-of-ordered-pair-x-y-where-x-and-y-are-positive-integers-less-than-30-such-that-x-2-2x-3-y-4-4y-5-7x-6-2

Question Number 118930 by Ar Brandon last updated on 20/Oct/20

Let [x] denote the greatest integer ⩽ x . Then the

number of ordered pair (x, y), where x and y are

positive integers less than 30 such that

[\frac{x}{2}] + [\frac{2 x}{3}] + [\frac{y}{4}] + [\frac{4 y}{5}] = \frac{7 x}{6} + \frac{21 y}{20}

is

(A) 1 (B) 2 (C) 3 (D) 4

Answered by floor(10²Eta[1]) last updated on 20/Oct/20

\frac{7 x}{6} + \frac{21 y}{20} \in N

\Rightarrow \frac{70 x + 63 y}{60} \in N

\Rightarrow 70 x + 63 y \equiv 10 x + 3 y \equiv 0 (\mod 60)

60 ∣ 10 x + 3 y

\Rightarrow 390 > 10 x + 3 y ⩾ 60

10 x + 3 y \in {60, 120, 180, 240, 300, 360}

10 x + 3 y = 60 z, 1 ⩽ z ⩽ 6

the solution to this diophantine equation is :

(x, y) = (6 z - 3 k, 10 k), k \in Z

\Rightarrow 0 < 10 k < 30 \Rightarrow k = 1 or k = 2

\Rightarrow y = 10

x \in {3, 9, 15, 21, 27}

y = 20

x \in {6, 12, 18, 24}

after checking that solutions on the

original equation the only one that works

are when y = 20 \Rightarrow x \in {6, 12, 18, 24}

Ans : D

Commented by Ar Brandon last updated on 21/Oct/20

Thank you Sir