Question

prove that gcd(p,q) +lcm of (p,q) -p-q is greater than or equal to 0 and gcd(p,q) +lcm of (p,q) -p-q is even​ (RMO question) where p and q are natural numbers

Answer 1

Step-by-step explanation:

gcd ( 100, 70 ) = gcd ( 170 , 70 )

100 = 2 x 2 x 5 x 5

70 = 2 x 5 x 7

=> gcd = 2 x 5 = 10 …………….. (1)

100 = 2 x 5 x 2 x 5

70 = 2 x 5 x 7

=> 100 +70 = {(2x5) x 2 x5} +{(2x5)x7 } = (2x5)(2x5 + 7)

& 70 = (2x5) x 7

=> gcd( 170, 70) = 2 x 5 = 10 .. . . . . . . (2)

This way, LHS = RHS

Now the PROOF:

p = a x b

q = a x c

=> gcd ( p, q) = a ………………… (1)

Now, p+ q = axb + axc = a ( b +c)

q = a x c

=> gcd ( p+ q, q) = a . . . . . . . . (2)

=> LHS = RHS