KVPY 2019 SB SX Exam Previous Papers

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Question : 11 of 120

Marks: +1, -0

The number of ordered pairs (a, b) of positive integers such that

{2a -1}/b

and

{2b -1}/a

are both integers is

1
2
3
more than 3

Solution:

{2a -1}/b≥1⇒a≥{b+1}/2⇒1/a≤2/{b+1}

⇒

{2b -1}/a≤{4b-2}/{b+1}=4-6/{b+1}<4

⇒

{2b -1}/a=1, 2, 3

2b – 1 is odd ⇒

{2b -1}/a=1,3

case (i) Let

{2b -1}/b=1

⇒

{2a -1}/b={2(2a -1)}/{a+1}=4-6/{a+1}

for a = 1,

{2a -1}/b= 4 -3 = 1

⇒a = 1, b = 1

for a = 3,

{2a -1}/b=4-3/2∉I

for a = 5,

{2a -1}/b=4 -1 = 3

⇒ a = 5, b = 3

case (ii) Let

{2a -1}/b=3⇒a = 3, b = 5

(similar as case (i))

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